Abstract
An axion insulator is a correlated topological phase, which is predicted to arise from the formation of a chargedensity wave in a Weyl semimetal^{1,2}—that is, a material in which electrons behave as massless chiral fermions. The accompanying sliding mode in the chargedensitywave phase—the phason—is an axion^{3,4} and is expected to cause anomalous magnetoelectric transport effects. However, this axionic chargedensity wave has not yet been experimentally detected. Here we report the observation of a large positive contribution to the magnetoconductance in the sliding mode of the chargedensitywave Weyl semimetal (TaSe_{4})_{2}I for collinear electric and magnetic fields. The positive contribution to the magnetoconductance originates from the anomalous axionic contribution of the chiral anomaly to the phason current, and is locked to the parallel alignment of the electric and magnetic fields. By rotating the magnetic field, we show that the angular dependence of the magnetoconductance is consistent with the anomalous transport of an axionic chargedensity wave. Our results show that it is possible to find experimental evidence for axions in strongly correlated topological condensed matter systems, which have so far been elusive in any other context.
Main
Axions are elementary particles that have long been known in quantum field theory^{3,4}, but have not yet been observed in nature. However, it has been recently understood that axions can emerge as collective electronic excitations in certain crystals, the socalled axion insulators^{5}. Despite being fully gapped to singleparticle excitations in the bulk and at the surface, an axion insulator is characterized by a quantum mechanical equation of motion, which includes a topological θE·B term, where E and B are the electric and magnetic fields inside the insulator and θ plays the role of the dynamical axion field. Physically, the average value of θ is determined by the microscopic details of the band structure of the system, and gives rise to unusual magnetoelectric response properties, such as quantum anomalous Hall conductivities^{6,7,8,9}, the quantized circular photogalvanic^{6,10,11} effect and the chiral magnetic effect^{6,12,13,14}. The prospect of realizing an axion insulator has inspired much theoretical and experimental work. Very recently, signatures of a dynamic axion field have been found on the surface of magnetically doped topological insulator thin films^{15,16,17}. However, the axionic quasiparticle in these systems—the axionic polariton^{5}—has so far eluded experimental detection. Alternatively, axion insulators have been predicted to arise in Weyl semimetals that are unstable towards the formation of a chargedensity wave (CDW)^{1,2,18,19,20,21}.
In their parent state, Weyl semimetals are materials in which lowenergy electronic quasiparticles behave as chiral relativistic fermions without rest mass, known as Weyl fermions^{22,23,24}. Weyl fermions exist at isolated crossing points of the conductance and valence bands—the socalled Weyl nodes—and their energy can be approximated with a linear dispersion relation (Fig. 1a). The Weyl nodes always occur in pairs of opposite ‘handedness’ or chirality. At low energies and in the absence of interactions, the chirality is a conserved quantum number and the two chiral populations do not mix. Parallel electric and magnetic fields (E‖B), however, enable a steady flow of quasiparticles between the left and righthanded nodes^{25}. This induced breaking of the chiral symmetry is a macroscopic manifestation of a quantum anomaly in relativistic field theory and gives rise to a positive longitudinal magnetoconductance in these systems. While the singleparticle states in Weyl systems have been well studied experimentally, manybody interaction effects remain largely unexplored. By turning on strong interactions, a CDW that links the two Weyl nodes and gaps the Weyl fermions can be induced in Weyl semimetals.
A CDW is the energetically preferred ground state of the strongly coupled electron–phonon system in certain quasionedimensional conductors at low temperatures^{26}. It is characterized by a gap in the singleparticle excitation spectrum and by a gapless collective mode formed by electron–hole pairs. The density of the electrons and the position of the lattice atoms are periodically modulated with a period larger than the original lattice constant (Fig. 1a). The phase ϕ of this condensate is of fundamental importance for experiments: its time derivative is related to the electric current density (j_{cdw} ≈ ∂ϕ/∂t) carried by the collective mode upon application of an electric field^{26}. Owing to its relation to the phase, this currentcarrying collective mode is called the phason. In most cases, the wavevectors of the CDW are incommensurate with the lattice, and the CDW is pinned to impurities. Therefore, only upon applying a certain threshold electric field E_{th} (above which the electric force overcomes the pinning forces) is the CDW depinned and free to ‘slide’ over the lattice^{27,28}, thereby contributing to the electrical conduction. The resulting conduction behaviour is strongly nonlinear and the electrical resistivity drops with increasing electric field.
In a Weyl semimetal, the CDW can couple electrons and holes with different chirality—that is, it generates a complex mass in the Dirac equation, which mixes the two chiral populations of the system even in zero magnetic field^{1,2,18,19,20}. The resulting gapped state is the axion insulator and the phase of the CDW is identified as the axion field (ϕ = θ). The phason is the axionic mode of this system and its dynamics is therefore described by the topological θE·B term. Compared with its highenergy version, this condensed matter realization of the axion has the advantage of being accessible in magnetoelectric transport experiments.
The pairing of electrons and holes of opposite chirality and the corresponding gapping of the Weyl cones induced by the CDW imply that the chiral anomaly is irrelevant for the singleparticle magnetoelectric transport in the axion insulator state. We find, however, that the axionic phason mode leads to a positive longitudinal contribution to the magnetoconductance. This connection can be understood through a calculation based on a phenomenological effectivefield theory for electrical conductivity (Methods). Applying E‖B to the Weyl–CDW system leads to an extra anomalous contribution from the chiral anomaly to the phason current, which can be identified as a signature of the axionic character of the CDW. Aligned E and B fields generate an additional collective chiral flow of charges with a rate that is proportional to E·B. The calculation yields a positive magnetoconductance contribution of σ_{CDW,xx}(B_{‖}) = c_{1} + c_{2}a_{χ}B_{‖} + c_{3} (a_{χ}B_{‖})^{2} in the axion insulator state, where c_{1}, c_{2} and c_{3} are materialspecific parameters, a_{χ} = 1/(4π^{2}) is the chiral anomaly coefficient and B_{‖} denotes the component of B that is parallel to E. We note that, owing to the threshold voltage caused by the impurity pinning in real material systems, c_{1,} c_{2} and c_{3}—and therefore σ_{CDW,xx}(B_{‖})—can in general depend on V (Methods). In the lowfield limit, where several Landau levels are occupied, the conductance is given by the quadratic term, recovering the characteristic, chiralanomalyinduced, positive longitudinal magnetoconductance of a noninteracting Weyl semimetal. In particular, in this limit we obtain a B_{‖}cos^{2}ϕ dependence of the magnetoconductance on the angle ϕ between E and B. In the highfield limit, the magnetoconductance behaves linearly with the magnetic field, similarly to the ultraquantum limit of a Weyl semimetal. This is the fingerprint of the axion, which allows us to probe the presence of the axionic phason mode through its effect on the electrical transport in a Weyl–CDW condensed matter system.
Recently, a positive magnetoconductance has been observed^{29} in the CDW phase of the predicted Weyl semimetal Y_{2}Ir_{2}O_{7} because of a reduced depinning threshold voltage in longitudinal magnetic fields. However, the phason current was found to be independent of B. To obtain evidence for the axionic phason, it is therefore desirable to go beyond these experiments.
In our study, we use the quasionedimensional material (TaSe_{4})_{2}I (Extended Data Fig. 1a), which has recently been shown to be a Weyl semimetal^{30} in its nondistorted structure (Fig. 1b). Its Fermi surface is entirely derived from Weyl cones with 24 pairs of Weyl nodes of opposite chirality close to the intrinsic Fermi level, E_{F} (10 meV < E − E_{F} < 15 meV). At temperatures below the transition temperature of T_{c} = 263 K, (TaSe_{4})_{2}I forms an incommensurate CDW phase, undergoing a Peierlslike transition^{31}, which is accompanied by the opening of a gap of approximately 260 meV to singleparticle excitations. In a recent study, the momentum (k) space positions of the electronic susceptibility peaks, calculated from the Fermi pockets of the Weyl points, and the k = (222) satellite reflections experimentally observed in XRD measurements of the CDW phase of (TaSe_{4})_{2}I have been found to match, indicating that the CDW nests the Weyl nodes and therefore breaks the chiral symmetry^{30}. In addition, (TaSe_{4})_{2}I has been previously shown to exhibit nonlinear electricaltransport characteristics^{32,33} in zero magnetic field upon crossing a threshold voltage V_{th}, which are associated with a currentcarrying sliding phason. Therefore, this material is a strong candidate for an axion insulator and can be used for the experimental observation of the axionic phason.
Crystals of (TaSe_{4})_{2}I grow in millimetrelong needles with aspect ratios of approximately 1:10, reflecting the onedimensional crystalline anisotropy of the material (Methods and Extended Data Fig. 1). We measured the electrical transport of five (TaSe_{4})_{2}I samples (A, B, C, D, E), with the electrical current I applied along the c axis of the crystals (Methods). In all samples, the lowbias fourterminal electrical resistivity ρ of the singlequasiparticle excitation was measured as a function of temperature T (Extended Data Fig. 1). The bias currentdependent transport properties were then measured for samples A, B and E. All investigated samples show similar electricaltransport properties. At 300 K, the ρ value of the crystals is around ρ_{0} = ρ(300 Κ) = 1,500 mΩ cm (±400 mΩ cm) with an electron density of n(300 Κ) = (1.7 ± 0.1) × 10^{21} cm^{−3} and a Hall mobility of μ(300 Κ) = (2.3 ± 0.1) cm^{2} V^{−1} s^{−1} (Methods and Extended Data Fig. 2), which are in good agreement with the literature^{32}. Here, ρ_{0} is the average value of all samples investigated and the uncertainty is the standard deviation. The uncertainties given for n and μ originate from the errors of the Hall fits. From further analysis of the Hall measurements (Methods), we find that the Fermi level in our samples is located precisely in the middle of the CDW gap and therefore at the position of the initial Weyl cones in (TaSe_{4})_{2}I (Extended Data Fig. 4). The data presented in the main text were obtained from measurements on sample A.
We first characterized the singleparticle transport at zero magnetic field (B = 0 T). Figure 1c (left axis) shows the ρ/ρ_{0} ratio of the singleparticle state on a logarithmic scale as a function of T^{−1}. For all samples investigated, we observe an increasing ρ with decreasing T, consistent with the ρ–T dependence of (TaSe_{4})_{2}I reported in the literature^{32,33}. The plot of the logarithmic derivative (Fig. 1c, right axis) versus the inverse temperature, T^{−1}, exhibits a well pronounced peak at around the expected CDW transition temperature, T_{c} = 263 K. The saturation of the logarithmic derivative at a constant value a_{v} above T_{c}^{−1} demonstrates that ρ/ρ_{0} follows the thermal activation law ρ/ρ_{0} ≈ exp[ΔE/(k_{B}T)], allowing us to deduce the size of the singleparticle gap ΔE in units of the Boltzmann constant, k_{B}. By averaging over the saturated range, we obtain a value of a_{v} = 0.72 ± 0.04 and thus ΔE = (259 ± 14) meV, which is in excellent agreement with the literature^{31,32}.
Next, we investigate the motion of the CDW at zero magnetic field in the pinning potential created by impurities. Figure 2a–c shows the variation of the measured voltage V as a function of the applied d.c. current I at four selected temperatures. Consistent with a sliding phason mode, nonlinearity in the V–I curves appears below T_{c} at high bias currents, that is, high voltages (Extended Data Fig. 5). Accordingly, the V–I curves can be well represented by a simple phenomenological model based on Bardeen’s tunnelling theory, as suggested by previous electricaltransport experiments on CDWs (Methods, Fig. 2d, Extended Data Fig. 6). To gain further insight into the collective charge transport mechanism, we calculate the differential electrical resistance dV/dI from the V–I curves and plot it versus the corresponding V for various values of T (Fig. 2e). For all temperatures investigated, dV/dI is constant at low V. However, below T_{c}, dV/dI decreases above a threshold voltage V_{th}, indicating the onset of the collective phason current^{26}. V_{th} is determined as the onset of the deviation from the zero baseline of the second derivatives d^{2}V/dI^{2} (see, for example, Extended Data Fig. 5). The magnitude of the corresponding E_{th} = V_{th}/L (L is the distance between the voltage probes) and its temperature dependence (Fig. 2f) are in agreement with the literature^{32}. Further analysis suggests that the nonlinear V−I characteristics are not the result of local Joule heating or a consequence of contact effects (Methods, Extended Data Fig. 6). These observations provide evidence for a propagating, currentcarrying CDW state.
We now test the axionic nature of the phason. For this purpose, we measure the nonlinear V−I characteristics at fixed T, but now in the presence of a background magnetic field. We apply a magnetic field between −9 T and +9 T in steps of 1 T, oriented perpendicular and in parallel to the direction of the current flow. All experimental observations are symmetric in B (compare Fig. 3 and Extended Data Fig. 7). As shown in Fig. 3a, b, no B modulation of the V−I characteristics and dV/dI can be seen within the measurement error of 0.1% for B perpendicular to I (E⊥B) across the whole magneticfield and temperature ranges investigated. To enable a direct comparison to the theory, we calculate the differential magnetoconductance Δ(dI/dV)_{B} = (dI/dV)_{B} − (dI/dV)_{0 T}. No Δ(dI/dV)_{B} is observed (Fig. 3c, d) in the perpendicularfield configuration. However, when the magnetic field is aligned with I (E‖B) (Fig. 3e, f)), the V−I characteristics and dV/dI display a strong dependence on the magnitude of the magnetic field B_{‖} above V_{th} and at low temperatures. Comparing the electrical currents at fixed T, B_{‖} and V, we find a strong enhancement of I with increasing B_{‖}. Because V_{th} itself is independent of B_{‖} in any direction up to ±9 T (Fig. 3b, f), the observed B dependence seems to originate from a fielddependent contribution to the phason current. As shown in Fig. 3g, h, we consistently observe a large positive Δ(dI/dV)_{B} for E‖B and T ≤ 155 K (see Extended Data Figs. 7, 8). The profile of this curve can be well described by a quadratic function below 7.5 T, according to the prediction for an axionic phason. To demonstrate the quadratic behaviour at low magnetic fields, we perform secondorder polynomial fits, \({d}_{1}{B}_{\parallel }+{d}_{2}{B}_{\parallel }^{2}\) (d_{1} and d_{2} are free parameters) to the experimental data and find that \({d}_{1}{B}_{\parallel }\ll {d}_{2}{B}_{\parallel }^{2}\) (Extended Data Fig. 9). By comparison to our theoretical transport model (Methods), the dominant quadratic behaviour for all B_{‖} < 7.5 T investigated is consistent with the estimated Fermi level in our samples near the position of the initial (undistorted) Weyl nodes. Above 7.5 T, Δ(dI/dV)_{B} deviates from the quadratic function, which is not yet understood. We note that Δ(dI/dV)_{B} in our experiment depends on V (Extended Data Fig. 9). As explained above, we attribute this observation to the impurity pinning of the CDW in our sample. The associated pinning potential induced strongly nonOhmic I–V characteristics, which are not included in our theoretical calculation.
To test the angular variation of the positive Δ(dI/dV)_{B}, we perform V–I measurements at 80 K and fixed 9 T for various angles φ (Fig. 4a). In accordance with the theoretical predictions for the axionic phason current, we find that Δ(dI/dV)_{φ} is described by a cos^{2}φ dependence. The observed pattern of angular dependence and the consistent functional dependence of the longitudinal Δ(dI/dV)_{B} on B are the fundamental signatures of the axionic phason and the associated chiral anomaly. This supports the identification of (TaSe_{4})_{2}I as an axion insulator.
In conclusion, our measurements reveal a positive longitudinal magnetoconductance in the sliding mode of the quasionedimensional CDW–Weyl semimetal (TaSe_{4})_{2}I, a signature that is linked to the presence of an axionic phason. We theoretically and experimentally find a \({B}_{\parallel }^{2}\) dependence of the longitudinal magnetoconductance and a cosinesquared dependence on the relative orientation of I and B. Our results show that it is possible to find experimental evidence for axions, which is particularly elusive in other systems, in strongly correlated topological condensed matter systems.
Methods
Phenomenological theory of the magnetotransport in an axionic CDW system
In this work, we model the magnetotransport in a CDW–Weyl system (that is, an axion insulator)^{1} using an effectivefield theory. For simplicity, we start with a quasionedimensional Weyl semimetal with exactly two Weyl nodes—although the real structure of (TaSe_{4})_{2}I is much more complicated. We assume that the Fermi surface consists of disconnected pockets surrounding these two Weyl nodes. Furthermore, we assume that the CDW ordering wavevector is commensurate with the Weyl node separation k_{q} and use natural units. To simplify the discussion, we neglect the effects of impurity pinning. We then discuss the effects of impurity pinning by examining the modifications to the B = 0 conductivity.
Effectivefield theory for the semiclassical limit, where many Landau levels are occupied
Our strategy is to introduce an order parameter
for the CDW order by decoupling the electron–phonon interaction Hamiltonian. Here \({c}_{({{\bf{k}}}_{{\bf{q}}}/2)+{\rm{\delta }}{\bf{k}}}\) is an operator that annihilates an electron displaced by wavevector δk from the Weyl node centred at ±k_{q}/2. By integrating out the electrons and neglecting the fluctuations in the gapped amplitude mode Δ, we arrive at an effectivefield theory for the phason mode ϕ of the CDW. Let us confine ourselves to zero temperature. By neglecting the chiral charge of the Weyl nodes (we will add it later), we find from Anderson^{34} the action (working in units where ħ = 1)
Here we have defined
with λ the electron–phonon coupling, and \({\omega }_{{{\bf{k}}}_{{\bf{q}}}}\)the phonon energy at wavevector k_{q}. N(0) is the density of states at the Fermi level, v_{F} is the Fermi velocity in the direction of the CDW wavevector, \({\hat{{\bf{k}}}}_{{\bf{q}}}\), and e* is the screened electron charge. The velocities c_{⊥} and c correspond to the transverse and longitudinal (phason) velocities, respectively. The final term in (2) corresponds to the sliding (Froelich) mode, excited by an electric field aligned with the CDW ordering vector.
Additionally, we find from the chiral anomaly the contribution^{34}
This contribution to the action emerges from the chiral anomaly in the undistorted Weyl semimetal. As shown in ref. ^{1}, the formation of the CDW causes the amplitude of the chiral anomaly to depend on the phason field. Let us now consider the effect of a uniform external magnetic field. Because we are interested in the response of the CDW to uniform fields, we restrict ourselves to spatially homogeneous configurations ϕ(x, t) = ϕ(t). By varying the action S = S_{0} + S_{c} with respect to ϕ, we find from the equations of motion that ϕ satisfies
To include the effects of damping, we follow Lee et al.^{35} and add a phenomenological damping term to this equation. Because we are interested in the dynamics of the sliding mode above the depinning transition, we do not consider a phenomenological pinning term. Defining an effective damping rate Γ, we thus find
We can solve this equation in the longtime limit to find the steadystate solution
Next, we look at the current density
carried by the CDW sliding mode, where A is the electromagnetic vector potential. By varying the action and restricting to positionindependent configurations of ϕ, we find
Finally, by combining equation (7) with equation (9), we find that
with the conductivity tensor given by
In particular, the magnetoconductivity is given by
We note that there are two types of contributions in equation (12). The first term depends on the relative orientation of the electric and magnetic fields relative to the CDW wavevector, and originates from a mixing between the chiralanomaly (B dependent) and conventional (B independent) contributions to the CDW dynamics in equations (7) and (9). The quadratic in B contribution to the magnetoconductivity, on the other hand, has a purely chiralanomaly origin. Although it corresponds to current carried by the sliding mode, it originates entirely from the underlying Weyl semimetal. Hence, like in the Weyl semimetal, this contribution to the magnetoconductivity is independent of the Weyl separation (that is, CDW ordering) vector.
We now note that in the idealized case in which the unordered phase is a perfect Weyl semimetal, N(0) → 0. In this case, the second term in equation (12) dominates over the first, and we recover the characteristic quadratic positive magnetoconductivity of a Weyl material^{36,37,38}. In particular, in this limit we also recover the characteristic cosinesquared dependence of the magnetoresistance on the angle between E and B. More generally, the quadratic term in equation (12) dominates when
Hence, we expect a positive quadratic magnetoconductivity in this regime.
We note that, unlike recent proposals for ‘axion insulators’ in Weyl CDW materials^{1,2}, our system preserves timereversal symmetry and hence has multiple pairs of Weyl fermions that are coupled in the CDW phase. This forces responses such as the anomalous Hall conductance to be zero. Nevertheless, the negative magnetoresistance due to the chiral phason channels is expected to persist in the limit of long phason lifetime (that is, weak intervalley scattering).
Microscopic argument for the quantum limit, where only the 0th Landau level is occupied
To gain some insight into how the sliding mode can influence the magnetoresistance in the quantum limit, we take our Weyl–CDW system and apply a large magnetic field along the \({\hat{{\bf{k}}}}_{{\bf{q}}}\) direction. We take the magnitude of B to be large enough so that only the zeroth Landau level is occupied when electron–phonon interactions are neglected. In this case, we have a truly quasionedimensional system, with conserved momentum \(Q={\bf{k}}\cdot {\hat{{\bf{k}}}}_{{\bf{q}}}\); each state is largely degenerate, with the degeneracy given by the total flux BA/(2π) through the system (measured in units of the flux quantum, with A the crosssectional area). We now reintroduce the electron–phonon interaction; this causes the formation of a CDW ground state. Because the Fermi points of the Weyl nodes disperse entirely in the lowest Landau level, the lowenergy physics of this transition matches that of a true onedimensional CDW transition, provided that we multiply all results by the Landau level degeneracy BA/(2π). In particular, Lee et al.^{28} showed that the sliding mode of the CDW contributes e^{2}n/(m*Γ) per channel to the electrical conductivity in the \({\hat{{\bf{k}}}}_{{\bf{q}}}\) direction, where m* = m(1 + m_{F}) is the electron effective mass, n is the onedimensional chargecarrier density and Γ is the phenomenological damping rate (phonon lifetime). By multiplying this by the number of channels we find in the quantum limit and for current and electric fields aligned with the magnetic field (and therefore CDW)
which is analogous to the magnetoresistance in the ultraquantum limit of a Weyl semimetal^{38}.
Comment on real material systems
Having justified the origin of the negative magnetoresistance in a simplified model, we now discuss how pinning effects will change the phenomenology. First, because CDWs are always pinned by impurities in real material systems, sliding only occurs upon reaching a certain threshold electric field, that is, the threshold voltage V_{th}. In addition, transport measurements always also involve singlecharge carriers thermally exited above the Peierls gap ΔE, which carry electrical current in parallel to the sliding CDW mode. Therefore, the B = 0 electrical conductivity (that is, the electrical resistivity) is in general nonOhmic and depends on the applied bias voltage V. Below the Peierls transition, the total electrical conductivity for a CDW system in zero magnetic field (σ_{ij,total}) is given by Bardeen’s tunnelling theory^{39}:
Here, σ_{SP} denotes the Ohmic conductivity contribution of the thermally exited singlecharge carriers, V_{th} is the threshold voltage and σ_{CDW,0} and V_{0} are voltageindependent parameters that usually depend on the temperature T. This is derived by assuming that pinning by impurities creates a gap of E_{g} < ΔE to CDW sliding. Equation (15) then follows from considering Zener tunnelling of the CDW across the gap. V_{0} is closely related to the dynamics of the sliding CDW and can be expressed as V_{0} = E_{g}/(2ξe*), where ξ is the coherence length, as in superconductors. Because E_{g} and e* do not depend on T, the temperature dependence of V_{0} provides a measure for the temperature dependence of ξ, which generally decreases with increasing T. σ_{CDW,0} is a measure of the number of carriers forming the CDW and has been observed to increase with increasing T. The validity of equation (15) for CDW systems has been confirmed by many experiments, by fitting the measured I–V characteristics to the current I_{total}(V) = I_{SP}(V) + I_{CDW}(V), corresponding to equation (15), with I_{SP}(V) = G_{sp}V accounting for the singleparticle contribution and I_{CDW}(V) = I_{0}(V − V_{th})exp(−V_{0}/V) for the sliding CDW. G_{sp} (the singleparticle electrical conductance), I_{0} (corresponding to σ_{CDW,0}) and V_{0} are the free fitting parameters. Importantly, there is no added conductivity below V_{th} and hence I_{total}(V) = I_{SP}(V) for V < V_{th}. V_{th} can be determined independently, as shown in Extended Data Fig. 5. By fitting I_{total}(V) to our data measured at zero magnetic field for V > V_{th} (Extended Data Fig. 5), we find excellent agreement with Bardeen’s tunnelling theory. The extracted value of V_{0} decreases and I_{0} increases with increasing temperature, in agreement with previous reports for other CDW systems. By fitting our measured I–V characteristics with the corresponding current expression I_{total}(V) = I_{SP}(V) + I_{CDW}(V), with I_{SP}(V) = G_{sp}V and I_{CDW}(V) = I_{0}(V − V_{T})exp(−V_{0}/V), using the singleparticle electrical conductance G_{sp} and I_{0} and V_{0} as the free fitting parameters, we find excellent agreement with Bardeen’s tunnelling theory (Fig. 2d, Extended Data Fig. 4).
Impurity pinning in real materials, such as in our samples, has important consequences on the experimental detection of the axionic response of CDWs. As mentioned above, the zerofield conductivity tensor will develop a nonlinear dependence on V. Additionally, the singleparticle current will always be measured in parallel to the sliding CDW in experiments. By putting these facts together, we see from equation (12) that the measured magnetoconductance \(\frac{\Delta I}{\Delta V(B)}\propto {\sigma }_{xx}(B){\sigma }_{xx}(0)\) will become nonOhmic and will in general depend on V. Hence, the amplitude of the characteristic cosinesquared dependence of the magnetoconductivity on the angle between E and B will also become Vdependent.
(TaSe_{4})_{2}I crystal structure and singlecrystal growth
(TaSe_{4})_{2}I is a quasionedimensional material with a bodycentred tetragonal lattice (Fig. 1b). The Ta atoms are surrounded by Se_{4} rectangles and form chains aligned along the c axis. These chains are separated by I^{−} ions. Single crystals of Ta_{2}Se_{8}I were obtained via a chemical vapour transport method using Ta, Se and I as starting materials^{40}. A mixture with composition Ta_{2}Se_{8}I was prepared and sealed in an evacuated quartz tube. The ampoule was inserted into a furnace with a temperature gradient of 500 to 400 °C with the educts in the hot zone. After two weeks, needlelike crystals had grown in the cold zone (Extended Data Fig. 1). Energydispersive Xray spectroscopy reveals a Ta:Se:I ratio of 17.75:72.95:9.3, which matches the theoretical ratio of 18.18:72.73:9.09 within the measurement precision of 0.5.
Electricaltransport measurements
All electricaltransport measurements were performed in a variabletemperaturecryostat (Quantum Design) equipped with a 9T magnet. To avoid contact resistance effects, only fourterminal measurements were carried out. To reduce any possible heating effects during current sweeps, the samples were covered in highly heatconducting, but electrically insulating, epoxy^{32}. The temperaturedependent electrical resistivity curves were measured with a standard lowfrequency (f = 6 Hz) lockin technique (Keithley SR 830), by applying a current of 100 nA across a 100MΩ shunt resistor. The applied bias current I = 100 nA for these measurements was chosen such that the corresponding electric field E = IR/L did not exceed the threshold field E_{th} > E, where L is the distance between the voltage probes. The determination of E_{th} is explained later. The bias currentdependent transport properties were then measured on samples A, B and E because their closely adjacent electrical contacts of L < 3 mm allowed us to cross the threshold field E_{th} with experimentally accessible currents (IR < 60 V) in our setup. The bias currentdependent transport experiments were limited to temperatures of T ≥ 80 K because E_{th} increases exponentially with decreasing T. Given the minimal contact resolution of >0.5 mm in our bulk samples, no switching could be achieved in our samples below 80 K, because the corresponding threshold voltages exceeded the measurable voltage range (<60 V). The d.c. voltage–current characteristics were measured with a Keithley 6220 current source (10^{14} Ω output impedance) and a Keithley 2182A/E multimeter.
Firstprinciples calculations
For the ab initio investigations we employ densityfunctional calculations (DFT) as implemented in the VASP package^{41}. We use a planewave basis set and the generalizedgradient approximation for a description of the exchangecorrelation potential^{42}. As a second step we create Wannier functions from the DFT states using Wannier90^{43} and extract the parameters for a tightbinding Hamiltonian to further analyse the band structure.
Estimate of the Fermi level position from Hall measurements
To estimate the Fermi level position in our samples, we performed Hall measurements on sample D. Because the singleparticle resistivity of all our samples was similar throughout the examined temperature range, the Hall measurements are representative of the whole set of samples presented here. The device used for the Hall measurements is shown in Extended Data Fig. 2a. Contacts 1 and 4 are used for electrical current injection, contacts 2 and 3 are used to probe the longitudinal voltage along the sample, and contacts 5 and 6 are used to measure the Hall voltage V_{5,6} across the sample. The magnetic field B is perpendicular to the measurement plane. The Hall resistance as a function of the magnetic field B is calculated using R_{H} = V_{5,6}/I at fixed temperatures T. We chose the electrical current I applied at each temperature to be small enough to probe only singleparticle transport but high enough to resolve the Hall signal V_{5,6}. In particular, we used I = 1 μA at 125 K, I = 10 μA at 155 K and 185 K, I = 100 μA at 215 K and 245 K, and I = 2 mA at 300 K. Although the data are quite noisy, we observe a strong B dependence of R_{H} (Extended Data Fig. 2c–h). Below 125 K, the longitudinal resistance of the sample became too high to measure the Hall response. As reported previously^{33}, the Hall resistance at low temperatures is linear with B. At 300 K, however, R_{H} becomes nonlinear with B at high fields, which is typical for semiconductors and insulators at elevated T, because both electrons and holes become thermally activated across the gap. At low temperatures, the band closer to the Fermi level dominates. As discussed previously, even in such a onedimensional material as (TaSe_{4})_{2}I, the Hall effect is well defined because, to the first order in B, the Lorentz force does not depend on the transverse effective mass^{33}. Recalling the standard expression for the lowfield Hall conductivity (singlechargecarrier approximation), we estimate the carrier concentration n = (dR_{H}/dB)^{−1}(e*d)^{−1} as a function of T from the slope dR_{H}/dB of the linear fits to the magnetic fielddependent R_{H} (d = 300 mm is the height of sample D). As seen in Extended Data Fig. 3i, n decreases by orders of magnitude when decreasing T from 300 K to 125 K, consistent with the increasing longitudinal singleparticle resistivity (Extended Data Fig. 2b). At the CDW transition temperature T_{c} a jump in n is observed, consistent with the opening of a singleparticle gap. At 300 K, the estimated carrier concentration is n = (1.7 ± 0.1) × 10^{21} cm^{−3}, leading to a mobility of μ = (2.3 ± 0.1) cm^{2} V^{−1} s^{−1}, using the Drude model μ = (e*nρ)^{−1} with an electrical resistivity of ρ = 1,562 μΩ cm at 300 K. Both n and μ are in good agreement with the values previously reported^{33} for (TaSe_{4})_{2}I.
Using the effective mass m* = 0.4m_{0} (m_{0} is the free electron mass) obtained from ARPES measurements^{32} for T < T_{c} and a singleband model, we subsequently determine the Fermi level position E_{F} in our sample with respect to the conduction band edge.
For a bandgap material, the carrier concentration is given by
where \({N}_{{\rm{c}}}=2{\left(\frac{2\pi {m}^{\ast }{k}_{{\rm{B}}}T}{{h}^{2}}\right)}^{3/2}\) and h is the Planck constant. By transforming equation (16), we obtain a description for E_{F} that only depends on T, m and n:
The result of equation (17) as a function of T is shown in Extended Data Fig. 2j. Within the error bar, the estimated E_{F} is independent of the temperature for T < T_{c}, giving a mean value of E_{F} = 129 ± 10 meV below the conduction band edge of the CDW gap. By comparing this number to the experimentally obtained CDW gap size of ΔE = 159 ± 5 meV, we find that the Fermi level in our (TaSe_{4})_{2}I samples is precisely in the middle of the CDW gap (E_{F} = ½ΔE). As shown previously via DFT calculations and ARPES measurements, the CDW gap opens symmetrically around the crossing points of the initial electronic band structure^{31}. Therefore, the Fermi level position is in the centre of the initial Weyl cones.
Excluding other intrinsic and extrinsic origins for the nonlinear V–I curves
The following analysis suggests that the nonlinear V–I characteristics are not the result of local Joule heating or a consequence of contact effects, but originate from an electrically driven intrinsic process in the (TaSe_{4})_{2}I crystals, consistent with a sliding CDW. First, we exclude thermally driven switching due to Joule heating. Thermally driven switching corresponds to a local rise of the temperature in the (TaSe_{4})_{2}I sample above T_{c}. Such a temperature rise would be determined by the power dissipated in the sample and the electronic circuitry. In the worstcase scenario, all of the V–I Joule heating power is dissipated within the (TaSe_{4})_{2}I crystals. Hence, the dissipated power necessary for the switching would approach zero as T → T_{c} from below. However, the electrical power P_{th} = V_{th}^{2}/[dV/dI(V_{th})] obtained at V_{th} from our experiments (Extended Data Fig. 6a) does not approach zero as T → T_{c}. Instead, P_{th} decreases with decreasing T. This is precisely the reverse of what we would expect from thermally driven switching. Second, we rule out contact effects as the cause of the nonlinear threshold behaviour in our experiments. As an example, we show V_{th} as a function of the distance of the voltage probes L at 80 K in Extended Data Fig. 6b for L = 1 mm, 2 mm and 3 mm. The observed linear V_{th}–L dependence implies that the transition is driven by the electric field and not by the absolute magnitude of I, which justifies our definition of E_{th} = V_{th}/L. It also demonstrates that the transition originates from a bulk effect, because the contacts in all of our devices are of identical size; hence, switching in the contacts would have no dependence on the length. Therefore, any change in the switching properties must result from the (TaSe_{4})_{2}I crystal itself.
Excluding other origins for the positive differential magnetoconductance
Here, we address the concern that the experimentally detected positive differential magnetoconductance Δ(dI/dV)_{B} in (TaSe_{4})_{2}I could arise from an alternative origin than the axionic nature of the CDW. Several intrinsic and extrinsic origins of the positive Δ(dI/dV)_{B} in (TaSe_{4})_{2}I can be excluded from the experimental observations and analysis. (i) Inhomogeneous current distribution caused by quenched disorder and the currentjetting effect are ruled out. The elongated geometry of the samples and the silver paint contacts used for current injection, encasing the entire ends of the wires, provide homogeneous current injection through the entire bulk of the samples^{44,45}.Theoretical simulations with similar sample geometries and even higher carrier mobility have shown^{44} that magnetic fields much larger than 9 T are required for the onset of currentjetting effects. To experimentally test for inhomogeneous current distribution in our wires, we use a special contact geometry on sample E (Extended Data Fig. 10). The voltage contacts on sample E were designed as point contacts, locally probing the voltage drop across two different edges of the sample. As in all other devices used in this study, the current injection contacts 1 and 6 cover the entire ends of the (TaSe_{4})_{2}I crystal. We measured simultaneously the potential difference V_{i–j} between the two pairs of nearestneighbour contacts along the current flow (V_{2–3} and V_{4–5}) at T = 80 K as a function of the applied current I and B for the perpendicular (I⊥B) and longitudinal (E‖B)I measurement configuration (Extended Data Fig. 10b–g, i–n) The obtained curves are nearly identical and display only small differences within the measurement error (up to 10%). The agreement across the two pairs of contacts shows that distortions of the current path are minimal and that the current distribution is uniform within the sample. This implies that the observed positive longitudinal Δ(dI/dV)_{B} is an intrinsic electronic effect. (ii) The absence of a Δ(dI/dV)_{B} for perpendicular B, as well as the large temperatures at which the longitudinal Δ(dI/dV)_{B} occurs, show that neither quantum interference effects of the condensate nor of the single quasiparticle excitations play an important role. (iii) The independence of V_{th} from B (Fig. 3a, e) excludes any possible effects from a reduction of V_{th}.
Data availability
All data generated or analysed during this study are available within the paper and its Extended Data files. Reasonable requests for further source data should be addressed to the corresponding author.
Change history
15 May 2020
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Acknowledgements
C.F. acknowledges the research grant DFGRSF (NI616 22/1; ‘Contribution of topological states to the thermoelectric properties of Weyl semimetals’) and SFB 1143. Firstprinciples calculations were funded by the US Department of Energy through grant number DESC0016239. B.A.B. acknowledges additional support from the US National Science Foundation EAGER through grant number NOAAWD1004957, Simons Investigator grants ONRN000141410330 and NSFMRSEC DMR1420541, the Packard Foundation and the Schmidt Fund for Innovative Research. Z.W. acknowledges support from the National ThousandYoungTalents Program, the CAS Pioneer Hundred Talents Program and the National Natural Science Foundation of China.
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B.A.B., C.F. and J.G. conceived the experiment. N.K., C. Shekhar and Y.Q. synthesized the singlecrystal bulk samples. J.G., S.H. and C. Schindler fabricated the electricaltransport devices. J.G. carried out the transport measurements with the help of S.S.H. and C. Schindler. J.G. and C. Shekhar analysed the data. J.N., Z.W. and Y.S. calculated the band structure. B.B. and B.A.B. provided the theoretical background of the work. All authors contributed to the interpretation of the data and to the writing of the manuscript.
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Extended data figures and tables
Extended Data Fig. 1 (TaSe_{4})_{2}I crystal structure, growth, device and transport characterization.
a, Crystal structure of (TaSe_{4})_{2}I. b, Sketch of the growth principle. A temperature gradient (temperatures T_{1} > T_{2}) is imposed on an evacuated quartz ampule, which contains (TaSe_{4})_{2}I powder at T_{1}. The evaporated (TaSe_{4})_{2}I diffuses towards the area with temperature T_{2} and condenses into single crystals. c, Optical micrograph of the asgrown (TaSe_{4})_{2}I crystals. d, Distribution of Weyl points in momentum (k) space of chirality ±χ. e, Scanning electron microscope image of a crystal. f, Typical device used for electricaltransport measurements. g, h, Singleparticle resistance of samples B, C, D and E. The electrical resistance R, normalized by R_{0} = R(300 K) (g) and its logarithmic derivative (h) as a function of T^{−1}, where T is the temperature. i, Singleparticle gaps of all (TaSe_{4})_{2}I samples investigated. The error denotes the fitting error of 1σ. The dotted line displays the mean value of all samples.
Extended Data Fig. 2 Hall measurements and Fermi level position of sample D.
a, Device used for the Hall measurements in a magnetic field B. b, Singleparticle longitudinal resistivity ρ versus temperature T. c–i, Singleparticle Hall resistance R_{H} at 125 K (c), 155 K (d), 185 K (e), 215 K (f), 245 K (g) and 300 K (h). The measured data (black) are fitted linearly (red) to extract the carrier concentration n (i). The error bars denote the error of the linear fits to 1σ. j, Estimated Fermi level position below T_{c}. ΔE is the singleparticle gap, obtained from Extended Data Fig. 1.
Extended Data Fig. 3 V–I characteristics of sample A at zero magnetic field and selected temperatures.
a–j, V–I characteristics at 80 K (a), 105 K (b), 130 K (c), 155 K (d), 180 K (e), 205 K (f), 230 K (g), 255 K (h), 280 K (i) and 305 K (j). These data were used to calculate the differential resistance dV/dI curves in Fig. 2d. At temperatures below 180 K, we start to observe nonlinearity. This nonlinearity becomes even more apparent in the dV/dI curves shown in Fig. 2d, where a deviation from the linear behaviour is already seen at 230 K.
Extended Data Fig. 4 Fitting the nonlinear V–I characteristics of sample A at zero magnetic field.
a–g, V–I characteristics (black line), a linear fit I_{sp}(V) = mV (green line) and a fit with the Bardeen model^{6} I(V) = I_{sp}(V) + I_{CDW}(V) (red line), where I_{CDW}(V) = I_{0}(V − V_{T})exp(−V_{0}/V) at 80 K (a), 105 K (b), 130 K (c), 155 K (d), 180 K (e), 205 K (f) and 230 K (g). h, i, m (h) and the threshold voltage V_{th} extracted from Fig. 2 were employed to extract the fit parameters. j, I_{0}. k, V_{0}. The error bars in h–k denote the error of the fits to 1σ.
Extended Data Fig. 5 Dependence of the switching voltage of sample A on the contact separation.
a, b, V–I characteristics (a) and differential resistance dV/dI (b) at 80 K and zero magnetic field B for various contact lengths L on sample A. c–e, Second derivatives d^{2}V/dI^{2} used to estimate the threshold voltages V_{th} shown in Fig. 2g. We define V_{th} as the voltage at which the last data point of d^{2}V/dI^{2} (marked by the vertical line) touches the zero baseline before the global minimum upon enhancing V.
Extended Data Fig. 6 Testing the origin of the nonlinear V–I curves of sample A.
a, The increasing Joule heating power P_{th} = V_{th}^{2}/[dV/dI(V_{th})] at the threshold, as a function of increasing T. b, The linear dependence of V_{th} on L shown at 80 K (the line denotes a linear fit) demonstrate that the observed effects are intrinsic to the (TaSe_{4})_{2}I crystals.
Extended Data Fig. 7 Symmetry and temperature dependence of the V–I characteristics in the magnetic field of sample A.
a, b, V–I characteristics (a) and differential resistance dV/dI (b) at 80 K for sample A in magnetic fields perpendicular to the applied current, but opposite to the magnetic field in Fig. 3. c, d, Corresponding V–I characteristics (c) and dV/dI (d) in magnetic fields parallel to the applied current. e–g, V–I characteristics at 80 K (e), 105 K (f) and 130 K (g).
Extended Data Fig. 8 Biasdependent data of samples A and B at 105 K.
The two samples have similar contact separation. a, b, V–I characteristics (a) and differential resistance dV/dI (b) at various magnetic fields B applied perpendicular to the current direction (I⊥B) for sample A. c, Magnetic field dependence of the magnetoconductance Δ(dI/dV)_{B} = (dI/dV)_{B} − (dI/dV)_{0 T} at 105 K and at various voltages V for sample A. d, e, Corresponding V–I characteristics (d) and dV/dI (e) at various magnetic fields B applied perpendicular to the current direction (I⊥B). f, Magnetic field dependence of the magnetoconductance Δ(dI/dV)_{B} for sample B.
Extended Data Fig. 9 Fitting parameters of the secondorder polynomial fits \({{\boldsymbol{d}}}_{{\bf{1}}}{{\boldsymbol{B}}}_{{\boldsymbol{\parallel }}}{\boldsymbol{+}}{{\boldsymbol{d}}}_{{\bf{2}}}{{\boldsymbol{B}}}_{{\boldsymbol{\parallel }}}^{{\bf{2}}}\) to the experimental longitudinal Δ(dI/dV)_{B‖} shown in Fig. 3g.
a, d_{1}. b, d_{2}.
Extended Data Fig. 10 Homogeneity test of the current distribution on sample E.
a, First configuration. I is injected from contacts 1 and 6 and V is measured between contacts 2 and 3. b–e, V–I characteristics (b, d) and dV/dI (c, e) at 80 K and at various magnetic fields B for I⊥B (b, c) and for I‖B (d, e). f, g, Δ(dI/dV)_{B} = (dI/dV)_{B} − (dI/dV)_{0 T} for I⊥B (f) and for I‖B (g). h, Second configuration. V is measured between contacts 4 and 5. i–n, Corresponding characteristics (i, k) and dV/dI (j, l) for I⊥B (i, j) and for I‖B (k, l). m, n, Δ(dI/dV)_{B} = (dI/dV)_{B} − (dI/dV)_{0 T} for I⊥B (m) and for I‖B (n).
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Gooth, J., Bradlyn, B., Honnali, S. et al. Axionic chargedensity wave in the Weyl semimetal (TaSe_{4})_{2}I. Nature 575, 315–319 (2019). https://doi.org/10.1038/s4158601916304
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