Abstract
The quantum Hall effect (QHE) is traditionally considered to be a purely twodimensional (2D) phenomenon. Recently, however, a threedimensional (3D) version of the QHE was reported in the Dirac semimetal ZrTe_{5}. It was proposed to arise from a magneticfielddriven Fermi surface instability, transforming the original 3D electron system into a stack of 2D sheets. Here, we report thermodynamic, spectroscopic, thermoelectric and charge transport measurements on such ZrTe_{5} samples. The measured properties: magnetization, ultrasound propagation, scanning tunneling spectroscopy, and Raman spectroscopy, show no signatures of a Fermi surface instability, consistent with infield single crystal Xray diffraction. Instead, a direct comparison of the experimental data with linear response calculations based on an effective 3D Dirac Hamiltonian suggests that the quasiquantization of the observed Hall response emerges from the interplay of the intrinsic properties of the ZrTe_{5} electronic structure and its Diractype semimetallic character.
Introduction
Electrons subject to a magnetic field B are forced to move on curved orbits with a discrete set of energy eigenvalues—the Landau levels (LLs). By increasing B, the LLs shift through the Fermi level E_{F} one after the other, leading to quantum oscillations in transport and thermodynamic quantities^{1}. At sufficiently large magnetic field, where only a few LLs are occupied, 2D electron systems (2DESs) enter the quantum Hall regime^{2,3,4,5}. This regime is characterized by a fully gapped electronic spectrum in the bulk and currentcarrying gapless edge states, leading to quantization of the Hall conductance G_{xy} = νe^{2}/h, where ν is the Landaulevel filling factor, e is the elementary charge, and h is Planck’s constant. The situation is different in three dimensions. Instead of fully gapping the bulk of the 3D electron gas, high magnetic fields confine the electron motion in the plane perpendicular to the magnetic field still allowing them to freely move along the field, making the electron motion onedimensional. Hence, current flow is allowed in the direction parallel to B. Crucially, the missing gap spoils quantization of G_{xy}. However, it is predicted that a 3D version of the QHE could occur in semimetals and doped semiconductors^{6,7,8,9}, in which the application of a magnetic field would lead to a Fermi surface instability, causing a periodic modulation of the electron density along the direction of B. Such modulation can effectively be thought of as a stack of 2DESs, each layer being in the quantum Hall regime. The signature of such a 3D quantum Hall system is that the Hall conductivity exhibits plateaus of σ_{xy} = νe^{2}/h·G_{z}/2π that are accompanied by a vanishing longitudinal electrical conductivity σ_{xx}^{6}. G_{z} is the reciprocal lattice vector of the modulation along B.
Recently, a quantized Hall response has been observed in the prototypical 3D Dirac semimetal^{10,11,12,13} materials ZrTe_{5}^{14} and HfTe_{5}^{15,16}. Particularly, the Hall resistivity ρ_{xy} has been found to exhibit a plateau, scaling with (e^{2}/h·k_{F,b}/π)^{−1} for the magnetic field aligned with the crystal’s baxis when only the last LL is occupied (quantum limit). k_{F,b} is the Fermi wave vector along the crystal baxis at zero magnetic field. The observed scaling of the plateau height with k_{F,b}/π has been interpreted as indicating a correlationdriven origin of the Hall effect. It was suggested^{14} that the applied magnetic field leads to a Fermi surface instability and formation of a chargedensity wave (CDW) with a wavelength of λ_{z} = π/k_{F,b} along the magnetic field. In a CDW, the density of the electrons and the position of the lattice atoms are periodically modulated with a wavelength, usually much larger than the original lattice constant^{17}. CDWs are usually the energetically preferred ground state of interacting quasionedimensional conductors due to the almost perfect nesting of the Fermi surface^{17}. In 3D systems, the dimensional reduction of the energy spectrum in high magnetic fields, in principle supports a scenario of a fieldinduced CDW, in which each CDW modulation “layer” contributes the conductance of one 2D quantum Hall system to the total bulk Hall conductivity.
The main argument for the CDW origin of the Hall plateaus in ref. ^{15} is the vanishing longitudinal resistance in the quantum limit, which was interpreted as a consequence of a fully gapped electronic structure in the bulk and nondissipative edge channels in each electron layer. However, the longitudinal resistivity seems to remain finite in the majority of the ZrTe_{5}^{14} and HfTe_{5}^{15,16} samples despite the presence of pronounced plateaus in the Hall resistivity (even down to 50 mK). Thus, it is desirable to investigate the physics of ZrTe_{5} in magnetic fields beyond simple charge transport experiments.
In transport, a CDW transition typically manifests as an abrupt increase of electrical resistance due to the gapping of the Fermi surface and nonohmic transport characteristics^{17}. However, a CDW ground state cannot be identified by transport measurements alone, as such features also exist in materials without a CDW transition^{18,19,20,21,22}. Both gapping of the Fermi surface and emergence of a periodic charge modulation should clearly impact thermodynamic, structural, and spectral properties^{1}. Gap openings in the electronic density of states (DOS) in nonmagnetic materials can, for example, be probed by magnetization measurements^{23}, thermoelectric coefficients^{24}, and scanning tunneling spectroscopy^{17}. The corresponding periodic charge modulation typically leads to a change in the phonon spectrum, which can be directly seen in Raman scattering^{17}, Xray diffraction^{25,26}, and ultrasound propagation measurements^{27,28}. In addition, Raman spectroscopy can directly probe the Raman active CDW amplitude mode^{17}.
Results
Experimental results
In our study, we have used similar ZrTe_{5} samples as those studied in ref. ^{14} grown by tellurium flux method and applied all the aftermentioned experimental techniques (see “Methods” for details). In total, we have investigated over 20 samples from which we exemplarily show data from Samples A, B, C, D, E, F, G, and H, all exhibiting consistent results (Supplementary Table 1). Upon cooling in B = 0 T, the longitudinal resistivity ρ_{xx} increases with decreasing temperature T until reaching a maximum at T_{L} = 90 K (Fig. 1a, Supplementary Fig. 1 and ref. ^{14}). This maximum has been previously observed in ZrTe_{5} and has been attributed to a Lifshitz transition that is accompanied by a change of chargecarrier type^{29}. Consistently, the sign of the zerofield Seebeck coefficient S_{xx} changes sign at T_{L}^{11,30}, indicating electrontype transport for T < T_{L} and holetype transport above.
Similar to previous studies^{14,16}, we find Shubnikovde Haas oscillations in ρ_{xx}(B) at low temperatures (T < 30 K) that are consistent with a single ellipsoidal Fermi (Fig. 1b). The magnetotransport measurement configuration is sketched in Fig. 1c. Further analysis of the Shubnikovde Haas oscillations reveal that the quantum limit is reached at magnetic fields as low as B_{QL} = 1.8 T for the magnetic field applied along the baxis (Supplementary Figs. 3, 6, 7, and 14) and a relatively small effective mass of the order of 0.01 m_{0} (Supplementary Table 1), which is in agreement with Dirac fermions (m_{0} is the bare electron mass)^{13}. Importantly, in this field configuration (see for example Fig. 1b), all studied samples show a pronounced plateau in ρ_{xy}(B) in the quantum limit with a height close to (h/e^{2}) π/k_{F,b} as reported in ref. ^{14}. However, ρ_{xx}(B) does not vanish in any of the measured samples, even when cooled to 50 mK^{16}_{,} consistent with the Landau quantization of a 3D Fermi pocket. While ρ_{xx}(B) remains finite in all samples, it is always much smaller than ρ_{xy}(B) at low temperatures and thus the Hall conductivity σ_{xy} = ρ_{xy}/(ρ_{xx}^{2} + ρ_{xy}^{2}) reduces to σ_{xy} ≈ 1/ρ_{xy}, enabling the observation of the quantization in ρ_{xy}.
To test the hypothesis of a fieldinduced CDW, we have first investigated S_{xx}(B) at low temperatures. S_{xx} is particularly sensitive to changes in the electronic DOS, i.e., gapping of the Fermi surface, because it is proportional to the energy derivative of the DOS at E_{F} (see Supplementary Information and ref. ^{31} for a detailed discussion)_{.} However, we do not find signatures of a CDW in the Bdependent S_{xx} with B applied along with the a, b, and c crystallographic axes and for various angles in between them. Instead, the data show quantum oscillations (Fig. 1d–f and Supplementary Figs. 4 and 9) that are consistent with a single ellipseshaped 3D electron pocket at E_{F} (Fig. 1g–i) as observed in ρ_{xx}(B).
Next, we have measured the magnetization of ZrTe_{5} across the hypothetical phase boundary proposed in ref. ^{14}. Magnetization is a thermodynamic quantity with the paramagnetic contribution directly proportional to the DOS and hence, highly sensitive to phase transitions^{23}, such as FS gapping due to the formation of a CDW (see Supplementary Note 6 for a detailed discussion). Figure 2a and b shows the temperature dependence of magnetization measured at 2 T and 100 mT with the field applied along the baxis—the field configuration in which the Hall plateaus are seen. The two investigated field values are chosen to cover two regimes: at 2 T the system is in the quantum limit and at 100 mT multiple LLs are occupied. In both regimes, we find that the magnetization does not show any signatures of the formation of a CDW. Investigation of the magnetization as a function of magnetic field (Fig. 2d and Supplementary Fig. 5) at low temperature reveals pronounced de Haasvan Alphen oscillations on top of a roughly linear background. The de Haasvan Alphen oscillations are in agreement with the Landau quantization of a single Fermi pocket and, hence, consistent with the observations in ρ_{xx}(B) and S_{xx}(B)^{32}. The strong linear diamagnetic background is indicative of a small effective mass, as expected for Dirac Fermions^{32,33}.
Independently from transport and thermodynamic probes, the gap opening in the electronic structure, related to the formation of a CDW, could be probed using scanning tunneling spectroscopy (STS), since STS provides direct information about the DOS as a function of energy. Therefore, STS would directly corroborate the presence of a fieldinduced gap tied to the formation of a CDW^{17}. We have performed STS experiments on our ZrTe_{5} samples at 0.4 K in fields up to 5 T applied along the crystallographic baxis (Fig. 2e and Supplementary Fig. 6). The measurements did not reveal any features in the density of states that could be attributed to a CDW formation. The DOS at E_{F} remained finite even deep in the quantum limit, consistent with the observed Landau quantization of a 3D Fermi surface in ρ_{xx}(B), S_{xx}(B), and M(B) and finite longitudinal resistivity. The observed band edges (labeled as CB_{1}, CB_{2}, and VB_{1} in Fig. 2e) are consistent with previous experiments on ZrTe_{5}^{34} and remain fixed with respect to each other—a characteristic feature of the lowest LLs of Dirac fermions^{35}.
In addition, we have performed ultrasound propagation and attenuation experiments. Measurements of the soundvelocity probe the system’s elastic modulus, and thus a thermodynamic quantity that is very sensitive to sudden changes in the free energy, such as those caused by a CDW gap opening^{27,28,36}. Ultrasound propagation measurements can provide information about the electron–phonon coupling, a quantity crucial for the formation of a CDW (see “Methods” and Supplementary Note 8 for a detailed discussion). Figure 2f and g shows the variation of the sound velocity Δv_{s}/v_{s} and the sound attenuation Δα of the longitudinal sound modes (propagation along the aaxis and longitudinal polarization vector along the aaxis) as a function of the magnetic field applied along the baxis at 2 K. We find that both Δv_{s}/v_{s} and Δα do not exhibit any anomalies that could indicate a phase transition. Instead, Δv_{s}/v_{s} and Δα consistently reflect the quantum oscillations observed in ρ_{xx}(B), S_{xx}(B), and M(B) that stand for a single Fermi pocket at E_{F} (see also Supplementary Fig. 5 for the transverse mode) It has been pointed out^{37} that for a CDW to emerge in the quantum limit regime above 1.5 T, the strength of the effective electron–electron interaction generated from electron–phonon coupling should be around g_{0} = 537.3 eV/nm. Our analysis of the quantum oscillations in Δv_{s}/v_{s,} however, suggests a much smaller coupling constant g_{0} = 0.015 eV/nm (see “Methods”). Since g_{0} depends quadratically on the electron–phonon coupling, we find that the electron–phononcoupling in ZrTe_{5} is too small by two orders of magnitude for a CDW to appear.
A fieldinduced phase transition into a CDW ground state is expected to lead to a modification of the crystal structure due to the new emerging charge modulation. To check if this is the case in ZrTe_{5} we have performed Xray diffraction measurements at zero field and in the quantum limit at 2 K. The formation of a CDW is expected to lead to the emergence of satellite superstructure peaks in the Xray spectra with corresponding kvectors of G_{z} = 2k_{F,b} adjacent to the main Bragg reflections^{17}. The results of Qscans along the bdirection of the crystal in the vicinity of the (010)Bragg peak with the field aligned along the baxis are shown in Fig. 3a. We find that the (010) Bragg peak does not change up to 2 T, and—most importantly—does not show any emerging satellite reflections^{26}. The XRay experiments indicate that the zerofield crystal structure of ZrTe_{5}^{38} is maintained in finite fields up to the quantum limit.
This result is confirmed by the magnetic fielddependent Raman spectroscopy measured at 2 K. The formation of CDW would cause additional peaks^{17,39} to appear in the Raman spectra of ZrTe_{5}^{40,41}. However, apart from small changes in the amplitude of the vibration modes, the Raman spectra do not change upon application of magnetic fields up to 9 T (Fig. 3b). In particular, the existing phonon modes do not show the change in Raman shift and no new spectral features appear that could be related to absorption by the Raman active CDW amplitude mode.
Our experimental investigations indicate that the state underlying the emergence of Hall plateaus in the quantum limit of ZrTe_{5} is in fact gapless and exhibits the behavior of a Fermi sea of a Landauquantized 3D Dirac semimetal.
Theoretical results
To directly compare the effect of Landau quantization of a 3D Dirac system on the Hall resistivity, we performed linear response calculations of the electrical transport properties, based on an anisotropic Dirac Hamiltonian, with a magnetic field along the zdirection (see “Methods”). As a crosscheck, we have also computed S_{xx,}, S_{xy,} Δv_{s}/v_{s}, and M (see “Methods” for details), using the same model. Although the inverse Landaulevel broadening and the transport relaxation time are in general sensitive to different scattering mechanisms, the calculations were performed assuming that both quantities are the same. This approximation has the advantages of being transparent to interpretation and that the calculated quantities can be directly related to the band structure. However, as a side effect, the calculated ρ_{xx} is slightly different than seen in the experiment (Fig. 4). More importantly, our model reproduces all other experimental data even on a quantitative level (Figs. 2 and 4) and explains the key observations of the electrical transport in ZrTe_{5}: first, at high magnetic fields, Landau quantization leads to plateaus in ρ_{xy}(B) with a height of (h/e^{2}) π/k_{F,b} in the quantum limit. For a 3D system, the Landau levels form continuous Landau bands dispersing parallel to the applied field. The Hall conductivity is the sum of conductance quanta over all occupied Landau bands v and wave numbers \({k}_{b}\), \({\sigma }_{xy}(B)=\mathop{\sum}\nolimits_{vocc.}{\int }_{{k}_{b}occ.}\frac{d{k}_{b}}{2\pi }\frac{{e}^{2}}{{\rm{h}}}=\mathop{\sum}\nolimits_{vocc.}\frac{2{k}_{F,b,v}(B)}{2\pi }\frac{{e}^{2}}{{\rm{h}}}\), where \({k}_{F,\nu ,b}(B)\) is the Fermi wave number parallel to the field. While \({k}_{F,b,v}\) in general depends on B, a characteristic feature of Dirac systems is that the lowest LL does not shift with magnetic field^{35} (see Supplementary Fig. 16) and \({\sigma }_{{xy}}\left(B\right)\) becomes \(\frac{2k_{F,b}}{2\pi }\frac{e^{2}}{\rm{h}}\) in the quantum limit. Second, while ρ_{xx}(B) shows Shubnikovde Haas oscillations with minima at the center of the Hall plateaus, it is finite for all magnetic fields, due to the gapless Landau band structure.
Magnetotransport in 2DESs with localized states is often described by effectively fixed chemical potentials, which leads to welldefined quantum Hall plateaus. In contrast, the large carrier densities n in ordinary 3D metals, such as copper, usually forces the particle number to be conserved in order to avoid large charging energies. This implies variation of the chemical potential as a function of magnetic field. In materials with small Fermi surfaces, such as Dirac semimetals, the charging energies on the contrary remain relatively small, since the absolute change in n with B at fixed E_{F} remains small. Defects can furthermore induce localized states that absorb some of the conduction electrons^{42}, and thermally activated higher bands may similarly serve as reservoirs of states that do not contribute to transport^{43}. As a result, the chargecarrier density in the conduction band can vary without the sacrifice of overall charge neutrality. In fact, the measured STS (Fig. 2e) shows that E_{F} does not shift with respect to the edge of the conduction band CB_{1}, i.e., lowest LL, up to 5 T. A variable particle number in the conduction band furthermore agrees with the observed Hall response that does not show the smooth behavior ρ_{xy} = B/ne (which would be expected if particle number were fixed). We find that the behavior of ZrTe_{5}, up to fields of several Tesla is in fact welldescribed by a model including only the itinerant electrons, but keeping E_{F} fixed.
Discussion
We now address the expected generality of our results. Our model provides a way to derive and understand the Hall conductivity of a genuine 3D electron systems from the conductance quantum scaled by a characteristic length when only the lowest few Landau bands are occupied. This length scale, however, does not relate to a 2D spatial confinement, but rather to an intrinsic momentum vector of the 3D electron system, which is given by the electronic band structure—a result that is also obtained for a stack of coupled quantum Hall layers^{16}. Therefore, we expect quasiquantized features in the Hall conductivity to be observed in the quantum limit of generic 3D metals and semimetals, regardless of the precise band structure, Fermi level, or purity as long as the particle number of conduction band electrons is not strictly preserved. In fact, our results are consistent with previous reports on ZrTe_{5} samples that exhibit different E_{F}^{44,45,46}. However, the shape of the quasiquantized features in σ_{xy} will depend on fine details of the 3D Landaulevel spectrum. Our findings promise to explain the often puzzling plateaus appearing in Hall measurements in many other low chargecarrier semimetals such as NbP^{47} or HgSe^{48} and in slightly doped semiconductors such as InAs^{42,49} and InSb^{50,51}, to name a few.
In summary, we have shown that ZrTe_{5} exhibits quasiquantized plateaus in the Hall resistivity that scale with \(\frac{2\pi }{2{k}_{F,b}}\frac{{\rm{h}}}{{e}^{2}}\) in the quantum limit due to an interplay of its Dirac band structure and low chargecarrier density. Our model can be directly applied to Landau quantization of generic 3D band structures, and also to the 3D anomalous Hall and SpinHall systems. Our findings establish the Hall effect in ZrTe_{5} as a truly threedimensional relative of the quantum Hall effect in 2D systems, and a prime candidate for the observation of relativistic chiral surface states^{18,19}.
Methods
Sample synthesis and preparation
Highquality singlecrystal ZrTe_{5} samples were synthesized with highpurity elements (99.9999% zirconium and 99.9999% tellurium), the needleshaped crystals (about (0.1 × 0.3 × 20) mm^{3}) were obtained by the tellurium flux method. The lattice parameters of the crystals were structurally confirmed by singlecrystal Xray diffraction. Prior to transport measurements, Pt contacts were sputter deposited on the sample surface to ensure low contact resistance. The contact geometry was defined using Al hard masks. Prior to Pt deposition, the sample surfaces were Argon etched and a 20nm Ti buffer layer was deposited to ensure good adhesion of the contacts. Deposition was conducted using a BESTEC UHV sputtering system. This procedure allowed us to achieve contact resistance of the order of 1–2 Ohm.
Sample environment
The pulsed magnetic field experiments up to 70 T were carried out at the Dresden High Magnetic Field Laboratory (HLD) at HZDR, a member of the European Magnetic Field Laboratory (EMFL). All transport measurements up to ±9 T were performed in a temperaturevariable cryostat (PPMS Dynacool, Quantum Design), equipped with a dilution refrigerator insert and a horizontal rotator.
Electrical and thermoelectric transport measurements
To avoid contact resistance, only fourterminal measurements were carried out. The longitudinal ρ_{xx} and Hall resistivity ρ_{xy} were measured in a Hallbar geometry with standard lockin technique (Zurich instruments MFLI and Stanford Research SR 830), with a frequency selected to ensure a phase shift below 1 degree—typically in the range of f = 10–1000 Hz across a 100 kΩ shunt resistor. In addition, some samples were measured, using a Keithley Deltamode resistance measurement setup for comparison. In both measurement modes, the electrical current was always applied along the aaxis of the crystal and never exceeded 10 μA in order to avoid selfheating. Thermoelectric measurements were performed using the same electrical contacts as for electrical transport measurements. In order to supply a substantial temperature gradient across the sample despite its high thermal conductivity, the sample was semisuspended with a heater attached to the freehanging end. In addition, a set of two Cernox Cx1030 thermometers were attached to the sample in order to measure the temperature gradient. In order to obtain thermoelectric data within the linear response regime, the applied gradient was kept at <10% of the sample temperature. The thermal voltage was measured using a Keithley 2182 A nanovoltmeter. In order to avoid the influence of parasitic thermal voltages on the cryostat cables, a background measurement without applied power was carried out both for Seebeck and Nernst measurements at all measured fields and temperatures.
Ultrasound propagation measurements
Ultrasound measurements in pulsed magnetic fields up to 10 Tesla were performed using a phasesensitive pulseecho technique. Two piezoelectric lithium niobate (LiNbO_{3}) resonance transducers were glued to opposite parallel surfaces of the sample to excite and detect acoustic waves. The transducer surfaces were polished using a focused Ion beam in order to ensure that the transducer attachment surfaces were smooth and parallel. The longitudinal and transverse acoustic waves were propagated along the aaxis with the transverse polarization vector along the caxis. Relative soundvelocity changes Δv/v, and sound attenuation Δα, were measured for field applied along the baxis. The longitudinal and transverse ultrasound propagation were measured at 28 and 313 MHz, respectively.
Magnetization measurements
Magnetization measurements were conducted in a standard Quantum Design VSM MPMS equipped with a 7 Tesla superconducting magnet. For measurements, the samples were attached on quartz sample holders and glued using a small amount of GEvarnish. In order to avoid parasitic contributions in magnetic measurements at small fields where the magnetic response of ZrTe_{5} is small, the background magnetization of the quartz holder together with the adhesive was measured and subtracted from the data.
Infield singlecrystal Xray diffraction
Infield singlecrystal Xray diffraction measurements have been performed at the Petra III P21.1 beamline at DESY (Hamburg, Germany). Measurements were performed in a standard cryostat equipped with a 10 Tesla horizontal superconducting magnet. For the measurements, the sample baxis was aligned along the field direction, caxis was in the scattering plane while aaxis was vertical. In order to detect new satellite peaks, several reciprocalspace directions were scanned both at 0 and 2 Tesla. Namely, we have performed kscans in the lowQ range [(0, 0, 0) – (0, 4, 0)] and, at highQ range [(0, 12, l) – (0, 16, l) with l = 0, 0.5, 0.75, 1, and (0, 16, l) – (0, 20, l) for l = 0, 1] where the Xray structure factor is expected to be stronger, For backgroundfree measurements, we have used a CdTe Amptek point detector with a combination of a Gegradient Si analyzer and a 101.7 keV incident energy beam.
Raman spectroscopy
ZrTe5 flakes were exfoliated by using the scotch tape method and transferred on a SiO_{2} substrate. The lowtemperature magnetoRaman measurements were performed using the attoRaman system and attoLIQUID 2000 cryostat system (Attocube systems AG, Germany) along with WITec 532nm laser and a WITec spectrometer (UHTS 300 SMFC, WITec, GmbH, Germany) equipped with 1800 g/mm grating and thermocooled CCD. The Raman spectra were integrated for 120 s and averaged over five accumulations.
Scanning tunneling microscopy
ZrTe_{5} crystals were cleaved in ultrahigh vacuum (base pressure ~1 × 10^{−10} mbar) at room temperature and immediately inserted into a cryogenic scanning tunneling microscope operated at T = 1.9 K. The relatively weak van der Waals bonding between the ZrTe_{5} layers offers a natural cleaving plane, resulting in a Teterminated surface. Scanning tunneling microscopy measurements were acquired using electrochemically etched W tips. Spectroscopic data were measured using a lockin technique, modulating the bias with 1 mV (r.m.s.) at 733 Hz.
Theory—the Dirac Hamiltonian
Abinitio calculations^{10} suggest ZrTe_{5} to be a Dirac semimetal whose lowenergy band structure at zero field can be modeled using an anisotropic Dirac Hamiltonian^{14}:
with \({\tau }_{i}\) (\({\sigma }_{i}\)) denoting Pauli matrices acting on the orbital (spin) degree of freedom and k_{j} and v_{j} denote the components of the momentum vector k and Fermi velocity in the jth direction, respectively. m accounts for the zerofield gap. The magnetic field \({\boldsymbol{B}}\,=\,{\rm{rot}}{\boldsymbol{A}}\) enters the Hamiltonian through the orbital effect which is implemented via the usual substitution: \({\hbar} {\boldsymbol{k}}\to {\hbar } {\boldsymbol{k}}+ e{\boldsymbol{A}}\), and the Zeeman effect is introduced via \({H}_{Z}=\frac{1}{2} g {\mu }_{B} {\tau }_{0} {\boldsymbol{\sigma }} \cdot {\boldsymbol{B}}\). Here, A is the vector potential, \(g\) is the Landé gfactor and µ_{B} is the Bohr magneton. The spectral function for the corresponding \(\nu\)th Landau band at b wave number \({k}_{b}\) is:
with the quantum lifetime \({\tau }_{Q}\). For the calculation of the electric and thermoelectric response functions, we have used the parameters: \(m=10\) meV, \(g=10\), \({v}_{a}=116392\frac{m}{s}\), \({v}_{b}=15340\frac{m}{s}\), \({v}_{c}=348875\frac{m}{s}\), \(\mu =12.745{\rm{meV}}\), \({{\rm{\tau }}}_{{\rm{Q}}}=3.064\) ps.
Linear response theory—electric and thermoelectric transport
Electric transport is calculated in linear response via the Kubo formula. The longitudinal and Hall conductivities are obtained from the respective currentcurrent correlation functions. The Green’s functions entering these equations contain an imaginary selfenergy to account for the scatteringinduced lifetime of quasiparticles. The thermoelectric conductivity tensor \(\hat{\epsilon }\) is obtained from the zerotemperature electrical conductivity tensor \(\hat{\sigma }\) via integration as \(\hat{\epsilon }=\frac{1}{\lefte\rightT}{\int }_{\infty }^{\infty }{\rm{d}}\epsilon \left(\epsilon \mu \right)\left(\frac{\partial {n}_{F}\left(\epsilon \mu \right)}{\partial \epsilon }\right)\hat{\sigma }\left(T=0,\mu =\epsilon \right).\) The thermopower tensor \(\hat{S}\), which contains the Seebeck and Nernst coefficients, is calculated via \(\hat{S}={\hat{\sigma }}^{1}\hat{\epsilon }\).
Partition function theory—magnetization
The magnetization M is calculated as the derivative of the free energy F with respect to the magnetic field, \(M=\frac{1}{V}\frac{{dF}}{{dB}}\), where V is the volume. The free energy is in turn defined via the canonical partition sum. Here we have neglected Landaulevel broadening (\({\tau }_{Q}\to \infty\)), but have otherwise used the same parameters as described above (see “Theory—the Dirac Hamiltonian”).
Theoretical model for phonon velocity renormalization
We consider a longitudinal acoustical phonon that propagates along the magnetic field direction (parallel to the \(b\)axis). In the long wavelength limit, the free phonon energy is given by \({\hbar}\Omega_{{\boldsymbol{q}}}={v}_{s}q\), where \({v}_{s}\) is the sound velocity and \(q\) is the phonon momentum. Electron–phonon interactions induce a selfenergy contribution \({\Pi }^{{\rm{R}}}({\boldsymbol{q}},\omega ={\Omega }_{{\boldsymbol{q}}})\) to the retarded phonon propagator. Its real part corresponds to the energy correction due to the electron–phonon interactions, \({\rm{Re}} \Pi^{{\rm{R}}}\left({\boldsymbol{q}},\omega = \Omega_{{\boldsymbol{q}}}\right)={\hbar} \Delta {\Omega }_{{\boldsymbol{q}}}= \Delta {v}_{s}q\)^{52}. Evaluating the selfenergy in a randomphase approximation at \(T=0\) and for finite Landaulevel broadening \(\Gamma \,> \, 0\) yields:
where \({\rho }_{\nu ,{k}_{b}}\)is the spectral function (see section 1.3) of the \(\nu\)th Landau band at b wave number \({k}_{b}\) and \(\lambda ={D}_{1}^{2}\lefte\right/(4{\pi }^{2}\rho {v}_{s}^{2}\hslash )=1.139\times {10}^{35}\frac{{Jm}}{T}\) is a fitting parameter. To account for samplespecific differences in the electronic band structure, we have chosen slightly different parameters in the effective Dirac Hamiltonian. Namely, we have changed the mass parameter to m = 9.5 meV, the gfactor to g = 7, the Fermi velocity bcomponent to v_{b} = 16486 m/s and the quantum lifetime to \(\tau_{Q}=0.658{\rm{ps}}\) in the calculation of the phonon velocity renormalization. The parameter \(\lambda\) is used to approximate the effective electron–electron interaction strength \(g_{0} = 4\pi^{2} \lambda v_{s} {\hslash}^{2} ((2 k_{F,b})^{2} + \kappa^{2})^{2} /  e \) for electrons at the Fermi level, \({k}_{z}=\pm {k}_{F,b}\), from the very recent work of Qin et al.^{37}. Here, \(\kappa =\sqrt{ \lefte\right^{3} B/(4 \pi^{2}\epsilon \hslash^{2} v_{b})}\) is the inverse Coulomb screening length with the dielelectric constant \(\epsilon /{\epsilon }_{0}=25.3\) evaluated at \(B=1{\rm{T}}\).
Data availability
All data generated or analyzed during this study are available within the paper and its Supplementary Information file. Reasonable requests for further source data should be addressed to the corresponding author.
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Acknowledgements
T.M. acknowledges funding by the Deutsche Forschungsgemeinschaft via the Emmy Noether Programme ME4844/11, the Collaborative Research Center SFB 1143, project A04, and the Cluster of Excellence on Complexity and Topology in Quantum Matter ct.qmat (EXC 2147). C.F. acknowledges the research grant DFGRSF (NI616 22/1): contribution of topological states to the thermoelectric properties of Weyl semimetals and SFB 1143. P.M.L., G.G., and Q.L. were supported by the US Department of Energy, Office of Basic Energy Science, Materials Sciences and Engineering Division, under contract DESC0012704. J.W. acknowledges support from the DFG through the WürzburgDresden Cluster of Excellence on Complexity and Topology in Quantum Matter ct:qmat (EXC 2147, projectid 39085490), the ANRDFG grant FermiNESt, and by HochfeldMagnetlabor Dresden (HLD) at HZDR, member of the European Magnetic Field Laboratory (EMFL). J.G. acknowledges support from the European Union’s Horizon 2020 research and innovation program under Grant Agreement ID 829044 “SCHINES”. We acknowledge DESY (Hamburg, Germany), a member of the Helmholtz Association HGF, for the provision of experimental facilities. Parts of this research were carried out at beamline P21.1 at PETRA III. Y.S. acknowledges funding from the Swedish Research Council (VR) with a Starting Grant (Dnr. 201705078) as well as Chalmers Area Of AdvanceMaterials Science.
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S.G and J.G. conceived the experiment. The single crystals were grown by G.G. The basic structural and transport properties of bulk crystals were measured and studied by P.M.L. and Q.L. S.G., R.W., A.M., and C.F. fabricated the final transport devices. S.G. and R.W. performed the transport experiments. S.G., M.K., D.G., S.Z., and J.W. performed ultrasound propagation measurements. S. D., F. K., and P.S. performed the Scanning Probe Spectroscopy. K. C. and A. S. carried out the Raman experiments. S.S.P.P. supervised the Scanning Probe Spectroscopy and the Raman experiments. S.G. and T.F. performed high field transport measurements. S.G., Y.S., M.Z., and O.I. prepared and executed the infield Xray diffraction experiments. S.G., M.B., P.S., and R.K. carried out the magnetization measurements. T.E. and T.M. provided the theoretical model of the threedimensional quantum Hall effect. S.G., T.M., T.F., and J.G. analyzed the data. All authors contributed to the interpretation of the data and to the writing of the manuscript.
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Galeski, S., Ehmcke, T., Wawrzyńczak, R. et al. Origin of the quasiquantized Hall effect in ZrTe_{5}. Nat Commun 12, 3197 (2021). https://doi.org/10.1038/s4146702123435y
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DOI: https://doi.org/10.1038/s4146702123435y
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