Magnetic field-induced non-linear transport in HfTe5

Abstract The interplay of electron correlations and topological phases gives rise to various exotic phenomena including fractionalization, excitonic instability and axionic excitation. Recently discovered transition-metal pentatellurides can reach the ultra-quantum limit in low magnetic fields and serve as good candidates for achieving such a combination. Here, we report evidence of density wave and metal-insulator transition in HfTe5 induced by intense magnetic fields. Using the non-linear transport technique, we detect a distinct non-linear conduction behavior in the longitudinal resistivity within the a–c plane, corresponding to the formation of a density wave induced by magnetic fields. In high fields, the onset of non-linear conduction in the Hall resistivity indicates an impurity-pinned magnetic freeze-out as the possible origin of the insulating behavior. These frozen electrons can be gradually reactivated into mobile states above a threshold of electric field. This experimental evidence calls for further investigation into the underlying mechanism of the bulk quantum Hall effect and field-induced phase transitions in pentatellurides.


Introduction
Topological phases of matter represent a wide range of electronic systems with nontrivial band topology.New quasiparticles, such as Dirac and Weyl fermions, forming by band crossing points in these topological matters, have been discovered in experiments 1- 5 .To date, most topological materials discovered so far fall in the scope of a single-particle picture. 6,7Owing to the vanishing density of states in the vicinity of band crossing points, the Coulomb interaction between these Dirac/Weyl fermions becomes unscreened with a long-range component, which may present distinct correlation effects 8 .One way to enhance the correlation is by using high magnetic fields to compress dilute electrons into highly degenerate states.By considering different interactions, various scenarios were proposed theoretically [9][10][11][12][13][14][15] in interacting topological semimetals.Among them, an important direction is to generate axionic dynamics by inducing density wave (DW) states 9 , which forms the quasiparticle of axions 16 , one of the most promising candidates for the dark matter.The pursuit of these proposals in experiments requests a low Fermi level so that electrons can be condensed into low-index Landau levels within the accessible field range.
Layered transition-metal pentatellurides ZrTe5 and HfTe5 are recently found to be close to an accidental Dirac semimetal phase in the boundary between strong and weak topological insulators, sensitively affected by the lattice constant [17][18][19][20][21][22] .While the exact value of the band gap is under debate, the bulk states of these systems can be regarded as Dirac fermions with a small mass term 19,23,24 .Angle-dependent quantum oscillations in ZrTe5 revealed a tiny Fermi surface, with a rod-like ellipsoid shape and large anisotropy in the Fermi wave vector and effective mass between the out-of-plane direction (b axis) and the in-plane directions (a and c axes) 20 .Owing to the easy access to the quantum limit within a moderate field, chiral anomaly 18,25,26 , log-periodic quantum oscillations [27][28][29] , anomalous Hall/Nernst effect 22,[30][31][32] , and quantized plateaus of Hall resistivity [33][34][35] have been observed in bulk crystals and flakes of ZrTe5 as well as its cousin HfTe5.These observations suggest pentatellurides as good candidates of topological systems for studying the electron correlation effect in magnetic fields.

Results
In this study, by combining linear and nonlinear quantum transport, we present evidences of field-induced DW and insulating states in HfTe5, which can be further modulated by an electrical field bias.Single crystals of HfTe5 were produced by the iodineassisted chemical vapor transport (CVT) as described in Section I of Supplemental Materials.HfTe5 has a carrier density of 2.7×10 17 cm -3 at 2 K, one order of magnitude lower than that of ZrTe5 (2.5×10 18 cm -3 ) grown by the same method 36 .The low carrier density enables an easy access to the quantum limit.We performed transport experiments with the current along the a axis and the magnetic field along the b axis noted as θ=0° configuration.Figure 1a shows the temperature-dependent resistivity with a peak around 72 K in Sample H1.This peak comes from the shift of Fermi energy from the valence band towards the conduction band as the temperature decreases.A similar transition also occurs in CVT-grown ZrTe5 as indicated by the sign change in both Hall and Seebeck coefficients with temperature 37 .Figure 1b shows the longitudinal magnetoresistivity (ρxx, the red curve) and Hall resistivity (ρxy, the blue curve) with clear oscillations at θ=0°.Apart from small quantum oscillations at low fields, ρxx experiences a large peak around 2 T when θ=0°, followed by a dip at 5.4 T. As marked by the Landau level index n at resistivity peaks, 3 / 13 HfTe5 enters the quantum limit around 2 T, where only the lowest zeroth Landau levels are occupied.The oscillation peak at n=1 reaches a much larger magnitude compared to the others at lower fields.Meanwhile, ρxy shows a series of plateau-like features as a function of B, in coordinate with the oscillations in ρxx.It resembles the bulk quantum Hall (QH) effect recently observed in ZrTe5 33,38,39 and HfTe5 34,35 but with a finite longitudinal resistivity residue.By tracking the angle dependence of quantum oscillations within the ab and c-b planes (Fig. S1 a-b), we can extract oscillation frequencies as functions of angles as shown in Fig. 1c (θ for the a-b plane and φ for the c-b plane as illustrated in the insets of Fig. 1c).It suggests that the Fermi surface of HfTe5 also adopts an ellipsoid shape (Fig. 1d), similar to that of ZrTe5.The fitting of the oscillation frequencies in Fig. 1c yields a strong anisotropy as ka: kb: kc=1:13.6:2.3, with ka, kb, and kc, the lengths of the three semiaxes in the Fermi ellipsoid, being 0.0045, 0.0613, and 0.0102 Å −1 , respectively.
We further investigate the transport properties of HfTe5 in higher magnetic fields.Figures 2 a and b show ρxx and ρxy of another HfTe5 crystal (Sample H2) measured in a resistive magnet up to 31.5 T at different angles.For θ=0°, the magnetoresistivity in the low-field regime well reproduces the results presented in Fig. 1 with a large dip in ρxx at ~4 T. Subsequently, ρxx continues increasing and gradually saturates above 21 T. No further oscillations appear since only the lowest Landau level is occupied.Notably, ρxy presents an anomalous peak-like feature (marked by the black arrows in Fig. 2b) at 11 T, then starts to decrease and finally saturates above ~21 T. As the field is tilted toward the in-plane direction, the magnetoresistivity ratio gets suppressed and the peak position (Bp) in ρxy moves towards higher magnetic fields.In Fig. S1d, we show that Bp can be fitted by the cosine function of the tilting angle θ, suggesting that the peak of ρxy is likely to be induced by the out-of-plane component of the magnetic field.Meanwhile, the plateau at ρxy=ρ1 persists at large θ as marked by the grey dashed line in Fig. 2b.The temperature dependences of ρxx and ρxy are plotted in Figs. 2 c and d, respectively.The suppression of ρxy in the high-field regime quickly disappears above 15 K and ρxx decreases drastically, while the low-field parts of ρxy overlap well at different temperatures.The field dependence of ρxx at 1.5~6 K shows a crossing point around 11.8 T (Fig. S6).It corresponds to a fieldinduced metal-insulator transition, which follows a scaling relation with T and B in the vicinity of the crossing point as shown in Section III of Supplementary Materials.The strong temperature dependence (1.5-15 K) of the high-field parts in ρxx and ρxy suggests that the peak in ρxy is unlikely to originate from the trivial multi-carrier transport mechanism.By considering the field dependence of both ρxx and ρxy, we conclude that HfTe5 enters an insulating state around 11.8 T, where ρxx is enhanced and ρxy is suppressed due to the decrease in the number of mobile carriers.
Strong nonlinear transport of HfTe5 is observed in magnetic fields.We started by applying a series of direct currents (DC) (0~1 mA) with a superimposed alternating current (AC) of 50 μA to the sample and detected the AC resistance using the standard lock-in technique.As presented in Figs. 2 e and f, the curves overlap for IDC=0 and 0.25 mA.As IDC exceeds 0.5 mA, the suppression of ρxy at high fields is gradually eliminated and ρxx decreases as well in the same regime.The bias-dependent resistivity indicates that the insulating phase of HfTe5 at high fields is suppressed upon large biases.
To track the evolution of the nonlinear behavior, we investigated the biasdependent DC differential resistivity.In Figs. 3 a and b, the differential resistivity (ρxx and ρxy) is plotted as a function of the biased electric field Eb above -15 T.Under small biases, the system remains in the linear transport regime with differential resistivity values fixed at different biases.Sharp transitions from linear to nonlinear regimes occur above a threshold electric field, ET.In the range of -15~-31 T, ρxx (ρxy) shows a prominent dip (peak) at ET followed by a linear decrease (increase) upon the increase of Eb.In contrast, in the range of -5~-10 T, ρxx increases with Eb beyond the threshold, while ρxy becomes nearly independent of Eb except for a slight variation near ET.It is in stark contrast to the conventional joule heating effect induced by large currents.The suppression of in-plane transport is gradually relieved by increasing Eb and the carriers become mobile again.The sharp transition near the onset of the nonlinearity will smear out as the temperature increases but the nonlinear transport persists (Fig. S5c).One unexpected finding is that ρxx continues to show nonlinear transport in the range of 0~-10 T (where the linear transport shows no anomaly) while ρxy does not (Figs. 3 c-d and Fig. S5a).At zero magnetic field, the nonlinear behavior vanishes (i.e., the resistivity becomes independent of the applied biases).
In Figs. 4 a-b, we extract the threshold electric field ET and the relative resistivity change Δρ/ρ near the transition as a function of magnetic fields.The relative resistivity change quantifies the effect on the transport property.The difference in nonlinear conduction of diagonal and off-diagonal components in resistivity tensors separates the phase diagram of HfTe5 under magnetic fields into two regimes.In Regime I, HfTe5 remains metallic and only diagonal resistivity from longitudinal transport is affected by bias fields (down-left inset in Fig. 4b).This behavior is consistent with the widely studied sliding motion of DW states.A bias exceeding ET results in a one-dimensional axial motion and does not contribute to Hall effect 40,41 .The differential resistivity curves shown in Fig. 3 are symmetric with electric fields and change linearly after the depinning transition, while other origins of nonlinear resistivity such as hopping conduction or p-n junctions give dramatic different behaviors (activation behavior in electric fields for hopping conduction 42 , nonsymmetric I-V curve for p-n junctions or other interface potential barriers 43 ).The sharp dip/peak features near the transition and the threshold electric field value are consistent with the typical sliding motion of density wave systems. 44,45,45,46In Regime II, electrons in HfTe5 becomes strongly localized and a large bias can activate both diagonal and offdiagonal resistivity tensors, making the system metallic again.Such behavior is distinct from the sliding DW case and fits the depinning process of defect-localized electron solid states.A large bias breaks the binding between electrons and impurities and makes electrons mobile again.Then both longitudinal and Hall transport (ρxx and ρxy) will be activated (top-right inset in Fig. 4b).We note that although a classical picture involving the "backflow" of normal electrons can affect Hall resistivity in the sliding DW state 47 , it leads to a decrease rather than an increase of |ρxy| and cannot give such a large change in Δρxy/ρxy, contradictory to our observation.

Discussion
In a previous study 33 , the emergence of bulk QH effect in ZrTe5 was interpreted as DW forming along the b direction induced by Landau level nesting.Similar to ZrTe5, HfTe5 shows signatures of QH effect 34,35 , which indicates that the interlayer dispersion is diminished.Nevertheless, the Landau level nesting picture cannot account for the in-plane nonlinear transport since it only induces DW along the b direction.The absence of the sliding behavior at zero field and the systematic increase of ET with B indicate that the DW state is induced by magnetic fields.The picture of field-induced spin DW as observed in 5 / 13 organic conductors 48 matches our results in a better way, while the charge DW order will normally be suppressed by the magnetic fields 49 .According to the quantized nesting model 48 , the strength of DW is governed by the Fermi surface anisotropy (refer to Section IV of Supplementary Materials for detailed discussions).Therefore, the interlayer periodic potential confines the system into a series of 2D planes, leading to the emergence of the bulk QH effect.Above 2 T, the carriers are confined to the zeroth Landau levels.The inplane transport can be evaluated via the bias-dependent differential resistivity measurements.Only a small part of carriers is localized by the DW at low fields as evidenced by the relatively small change of the resistivity between the pinning and sliding states.This conclusion also explains another unusual behavior -the sign of Δρxx/ρxx oscillating with quantum oscillations at low fields shown in Fig. S5a.Both the normal and DW electrons coexist in the system with the former being the majority.Under large biases beyond ET, the DW electrons start one-dimensionally sliding motion and contribute to the conduction.Then according to the two-fluid model, the current of normal electrons becomes smaller since the total current is fixed.Therefore, the quantum oscillations given by normal electron conduction will become comparably weaker, which results in the oscillating sign of Δρxx/ρxx with B as shown in Fig. S5a.
Based on these observations, we propose a possible scenario for the phase diagram of HfTe5 in magnetic fields.The gap size of HfTe5 decreases with temperature, resulting in the low-temperature phase close to an accidental Dirac semimetal. 19,21At zero field, the Fermi surface of HfTe5 resembles a highly anisotropic ellipsoid filled with electrons.With magnetic fields along the b direction, the system is firstly driven to a three-dimensional density wave state, then gradually collapses into an impurity-pinned insulating state around 11.8 T. Generally, there are mainly two kinds of mechanisms accounting for the defectpinned insulating phase in the quantum limit of dilute electronic systems.One is the formation of Wigner crystals 50 .It is a collective state where the potential energy of electrons dominates the kinetic energy and the electrons crystallize into a lattice.However, the typical temperature required to form a Wigner crystal is in the milli-Kelvin range while the insulating phase in HfTe5 persists up to over 15 K. Besides, the critical field of Wigner transition was found to be temperature dependent 51 while the transition features in HfTe5 were found to be temperature insensitive.The other one is the magnetic freeze-out effect, which originates from the electron-impurity interaction 52 .In this case, electrons bind to impurities due to the small magnetic length at high fields (Fig. 4c).In Section IV of Supplementary Materials, a rough estimation based on the effective mass and the carrier density in HfTe5 gives the value field of the magnetic freeze-out Bc≈8.2 T, close to the experimental results.In the view of band structure, the Fermi level meets the impurity band at high magnetic fields and electrons are localized near the impurity sites as illustrated in Fig. 4d.Thus, the Hall conductivity should be thermally activated by temperature as   ∝  −  / with   the field-dependent binding energy and  the Boltzmann constant.As shown in Fig. S6, the absolute value of the Hall conductivity increases with the temperature above 14 T. Figure 4e shows the activation of   in different fields.The fitted activation gap Δ as a function of the square root of the magnetic field is plotted in the inset of Fig. 4e.It increases linearly with √ in the range of 14.5~20 T and gradually saturates to ~0.4 meV after that, in agreement with the magnetic freeze-out case as detailed discussed in Section IV of Supplementary Materials.
One interesting feature is that ρxx only changes by less than 40% at different biases while ρxy is suppressed to nearly zero in the high-field localized states.In Fig. S7, we show that in high fields the Hall component of conductivity gradually exhibits different temperature dependence from the longitudinal part.While further investigations are on demand, it may be related to possible 1D edge modes as proposed recently in KHgSb 53 .
The observations of in-plane nonlinear conduction and possible magnetic freeze-out state indicate a complicated phase diagram of HfTe5 in magnetic fields.Thermodynamic properties and the ac transport of sliding DW may be investigated to further identify the nature of these phase transitions.And experimental methods sensitive to spin-polarization can be applied to settle down whether it is a charge or spin DW.Meanwhile, the unique electromagnetic response (axion electrodynamics) in topological systems allows for the fluctuation of topological θ-term in the presence of the DW order, which behaves like the axion and gives rise to novel topologically protected transport properties. 9,54The DW state of HfTe5 in magnetic fields can be therefore used to explore the physics of axionic dynamics 9,12,54 .

Methods
High-quality single crystals of HfTe5 were grown via chemical vapor transport with iodine as the transport agent, similar to ZrTe5 20 .Stoichiometric Hafnium flakes (99.98%,Aladdin) and Tellurium powder (99.999%,Alfa Aesar) were ground together and sealed in an evacuated quartz tube with iodine flakes (99.995%,Alfa Aesar).A temperature gradient of 60 ℃ between 490 and 430 ℃ in a two-zone furnace was used for crystal growth.The typical as-grown sample has a long ribbon-like shape along a-axis.The sample crystalline quality and stoichiometry were checked by the X-ray diffraction and the energy dispersive spectrum.The magneto-transport measurements were performed in a Physical Property Measurement System (Quantum Design) for the low fields and in resistive magnets in Tallahassee, US and Hefei, China for the high fields with the standard lock-in technique.The differential resistivity measurements were carried out using the corresponding in-build mode of Keithley 6221 and 2182 models.The bias electric field is extracted as   = / with ,  and  being the applied current, four-terminal resistance value, and the channel length between two inner voltage-sensing electrodes, respectively.

Figure 1 .
Figure 1.Magneto-transport and Fermi surface anisotropy of HfTe5.(a) The zero-field resistivity of HfTe5 as a function of T with a peak at 72 K.The current is applied along the a direction within the a-c plane.(b) The longitudinal (ρxx, the red curve) and Hall (ρxy, the blue curve) resistivity at 2 K with n the Landau level index.Note that the original longitudinal and Hall resistance data of this figure (without normalization with sample geometry) has been presented in Fig. S18d of Ref. 36 by part of the authors.The dashed lines mark the position of Hall plateaus.(c) The oscillation frequencies as a function of θ and φ.The insets are the measurement geometries.(d) The sketch of the Fermi surface in HfTe5.

13 Figure 2 .
Figure 2. Linear and nonlinear transport results of HfTe5 at high magnetic fields.(a-b) ρxx (a) and ρxy (b) as a function of B at different angles at 1.5 K.The inset of a is the measurement geometry.The intervals between curves in the ranges θ=0°~ 60° and θ=66°~90° are 6° and 3°, respectively.The transition in Hall effect is marked by the black arrows.(c-d) ρxx (c) and ρxy (d) at different temperatures at θ=0°.The intervals between curves in the ranges 1.5~6 K, 6~12 K and 15~40 K are 0.5 K, 1 K and 5 K, respectively.The crossing point of ρxx and the transition in ρxy are marked by the black arrows.(e-f) ρxx (e) and ρxy (f) at different DC currents at θ=0° and 1.5 K.The inset of e is the measurement configuration.Each color represents the same θ, temperature, current for a and b, c and d, e and f, respectively.Note that the field-dependent magnetoresistivity data in this figure have been symmetrized (or anti-symmetrized for Hall effect) in B with the original data shown in Fig. S4.

13 Figure 3 .
Figure 3. Bias-dependence of resistivity tensors at different magnetic fields.(a-b) ρxx (a) and ρxy (b) as a function of Eb fields at 1.5 K.The interval between curves is 1 T. (c-d) ρxx (c) and ρxy (d) as a function of Eb at 1.5 K.The interval between curves is 1 T. The value of Eb=1 V/cm corresponds to a DC current of 1.09 mA at 31 T.

Figure 4 .
Figure 4.The DW and insulating phases of HfTe5 in magnetic fields.(a-b) ET (a) and Δρ/ρ (b) as a function of B. The navy blue and purple dots correspond to ρxx and ρxy, respectively.Δρ/ρ is the resistivity difference Δρ before and after the transition (Fig. 3b) divided by the zero-Eb value of the resistivity.The insets in a show the nonlinear behavior in ρxy, which separates the phase diagram of HfTe5 into two regimes.In Regime I, only the longitudinal transport will be activated.In Regime II, both longitudinal and Hall transport will be activated when the electric field breaks the binding between electrons and impurities.The inset in b is a schematic of the DW electron-sliding.(c) The magnetic freeze-out state where the carriers bind to the local impurities (the grey circle).(d) The schematic band structure for the magnetic freeze-out effect.At high fields the itinerant electrons are pinned to the impurity band.(e) Arrhenius plots of |σxy|.The inset is the activation gap as a function of B 0.5 .