Introduction

The low-energy states of electrons in topological materials can be described as a series of quasiparticles, which obey different representations of the Dirac equation1,2,3,4,5. One kind of the three-dimensional (3D) massless quasiparticle is Weyl fermion, which has been discovered in topological Weyl and Dirac semimetals6,7,8,9,10,11,12,13. The Weyl quasiparticles occur in the vicinity of a finite number of band touching points, dubbed Weyl nodes, in these topological semimetals. The unique topological nature of the Weyl semimetal promises many novel properties belonging to the massless quasiparticles, such as linear energy dispersion, monopoles, and Fermi arcs on the surface14,15,16. Of particular, the 0th Landau bands (LBs) of the Weyl fermions in strong magnetic field are purely chiral modes17,18,19. These one-dimensional (1D), chiral LBs are expected to exhibit a negative longitudinal magnetoresistance (MR) as a signature of the long-sought chiral anomaly in quantum field theory20. Realizing the chiral anomaly in solids has inspired intensive experimental activities on topological semimetals in strong magnetic field21,22,23,24,25,26,27. Nevertheless, the MR in the quantum limit (QL) also depends sophisticatedly on the nature of the impurity scattering28 and thus it cannot give the information of the quasiparticles’ spectrum deterministically29,30,31. Actually the alleged linear positive longitudinal magneto-conductance in theory has not been observed experimentally in the QL of topological semimetals.

By contrast, the magnetic responses of the electrons are much less complex, because they do not interplay with impurity scattering. Indeed they are simply determined by the derivatives of the electrons’ thermodynamical potential Ω with respect to magnetic field H, and they have been used to probe the properties of the Fermi surface, including the relativistic aspects32,33,34. Previous studies claimed that the non-trivial Berry phase accounts for the non-zero extrapolation of the quantum oscillations (QOs) in topological semimetals28,35,36,37,38,39. This Landau-fan diagram-based criterion is often blurred by complicated resistivity tensors28. By contrast, we suggest a method for detecting relativistic quasiparticles by using of the magnetic responses in this paper. We focus on the magnetic response of the 0th LB in a sufficiently strong magnetic field when all the rest LBs have left the Fermi surface. In such extreme condition, the crossing bands show sheerly different magnetic response than that of trivial parabolic bands. To illustrate the difference, we choose TaAs as a prototype topological semimetal hosting well-defined Weyl quasi-electrons. The cross section area of the Weyl pocket is sufficiently small so that a steady magnetic field can approach the QL within a large deviation of the angle in a magnetic torque measurement. We show that the effective transverse (MT) and parallel magnetizations (M||) of TaAs are quasi-linear field dependent beyond the QL. Consistent with our calculations, these non-saturating magnetic responses are distinct from that for non-relativistic quasiparticles. Our observation serves as a thermodynamic method for discerning the relativistic quasiparticles in emergent topological materials.

Results

Theory of high-field magnetization

Before discussing the experimental results, we first compare the pictures of the magnetic response for the relativistic and non-relativistic electron systems (Supplementary Note 2). Figure 1a, b is the sketches of the bands for the non-relativistic electrons and holes in a magnetic field, in which the magnetization is contributed by both conduction and valence bands, which are separated by a band gap. To calculate the full-response magnetization, we include the contributions of the valence bands and conduction bands. When the field increases, the LBs successively leave the Fermi surface (Fig. 1b), leading to the oscillatory M|| around zero and the dropping MT with increasing field (Fig. 1e, f) until all of the LBs have left the Fermi surface except for the lowest one. The Fermi energy EF almost remains intact in low fields when there are a lot of LBs below it, so the low-field simulations for a fixed EF and for a fixed carrier concentration (Nc) are almost identical. However, the change of EF in strong magnetic fields cannot be ignored when the system enters the QL (Supplementary Note 2). Here we consider two possible constraints for the calculations: imposing the conservation of Nc or fixing EF. If EF is fixed, the zero-point energy will lift the 0th LB above EF beyond a critical magnetic field, leading to vanishing M|| and MT (red dot lines in Fig. 1e, f). If Nc is fixed, EF will be pinned at the edge of the 0th LB regardless of the increase of the magnetic field. Under this constraint, M|| will saturate and MT will be a constant (blue lines in Fig. 1e, f). This high-field behavior is obvious in Supplementary Eqs. (10) and (62), where both the contributions from valance and conduction bands saturate at high fields. In other words, both M|| and MT for the non-relativistic electrons are invariant in the QL due to non-relativistic band dispersion. This picture is tenable for an artificial case where two parabolic bands touch at a point, as shown in Fig. 1c, d. Our calculations prove that the magnetization behaviors of this case in the QL is similar to that with a gap (Supplementary Note 2). The profile of the saturating M|| and MT in the QL has been observed in 3D massive (gapped) bulk systems, such as Bi, sulfur-doped Bi2Te3, and InSb34,40,41.

Fig. 1
figure 1

Magnetic responses of the non-relativistic and relativistic fermions. a The energy bands of non-relativistic (parabolic-band) fermions in zero magnetic field. b The LBs of non-relativistic fermions form in a magnetic field. Only the 0th LB crosses the EF in the QL. c Two energy bands of non-relativistic (parabolic-band) fermions touch each other in a momentum point in zero magnetic field. d The LBs of non-relativistic fermions touch each other in a magnetic field. e, f Calculated parallel magnetization (M||) and effective transverse magnetization (MT) of non-relativistic fermions (the two cases show identical behaviors in the QL, here we just show a representative one for brevity) as functions of magnetic field. We used two constraints for the calculation in the QL: fixed the EF (red line) and fixed carrier density (Nc) (blue line). g A typical schematic energy band of a pair of type I Weyl nodes in zero magnetic field. h A series of LBs for a pair of Weyl nodes form in a magnetic field. Two 0th LBs are entirely chiral (green and violet). i A typical relativistic fermion band (Dirac model) with a small gap. The gap is much smaller than the cutoff energy Λ. This model also gives out the same non-saturating behavior as we see in the Weyl case. j The corresponding series of LBs for a simple Dirac model with a small gap in a magnetic field. k, l Calculated M|| and MT as functions of magnetic field, respectively. The two cases show identical behaviors in the QL, and here we just show a representative one for brevity

Full size image

By contrast, gapless Weyl fermions contribute to the magnetization in a different way due to the band inversion and relativistic spectrum (Fig. 1g, h). Both M|| and MT oscillate around a mean value in low magnetic fields (Fig. 1k, l). When all the N > 0 LBs have left EF, both chiral modes contribute to the magnetic response (Fig. 1h), leading to non-saturating M|| and MT in strong magnetic fields. We then consider a generalized case in which the Dirac model with a mass term is adopted, instead of the massless Weyl model (Fig. 1i, j). We are interested in whether this behavior still exists for a generalized model, namely the Dirac model with a mass term. The simulations for that model show a similar and consistent result compared with that of the Weyl case only when the gap is much smaller than the cutoff energy (Λ). In the large-gap limit, it is natural to expect that the model reduces to a trivial case as shown in Fig. 1a. This means that the relativistic feature, e.g., the linearity of the energy dispersion, should present in the Dirac model and thereupon the non-saturating magnetization in the QL is expected. The essential difference of the magnetic responses in the QL stems from the nature of relativistic electrons. In the following theoretical part, the origin of this difference will be discussed in detail, by including both the conduction and valence Weyl bands.

Magnetic torque and effective transverse magnetization

The depiction above sheds light for understanding the magnetic torque signals of TaAs. The magnetic torque τ is defined as \(- \frac{{\partial {\mathrm{\Omega }}}}{{\partial \theta }}\), where θ is the angle of the magnetic field with respect to the c-axis. In a macroscopic expression τ is formulated as a mechanic torsional torque, τ = VM × μ0H = μ0VHMT. Figure 2a, b shows τ and MT against magnetic field at several representative tilted angles, respectively. Clear de Haas–van Alphen (dHvA) QOs superpose on a large diamagnetic background at different temperatures. The dHvA oscillations are predicted by Landau’s theory for diamagnetism, and this Landau diamagnetism is small in metals and usually smeared by other magnetic signals due to the spin contribution (Pauli paramagnetism) and the core shell (Lamor diamagnetism). However, the electron mass in semimetals is renormalized by the Fermi liquid theory and the orbital contribution of the Landau diamagnetism gains a huge enhancement entailing a square relation42. In TaAs, the cyclotron mass m* is 0.06 m, where m is the free electron mass (Supplementary Note 3), and therefore the enhancement takes the form of \(\frac{{\chi _{{\mathrm{Landau}}}}}{{\chi _{{\mathrm{Pauli}}}}}\sim (m/m^ \ast )^2\)42 leading to a value about 300. Bearing in mind that the Lamor diamagnetization and Pauli paramagnetization have the same order, we believe that the magnetization in TaAs is dominated by the Landau diamagnetism in the whole range of magnetic field. In essence, only one frequency is detected at small angles and no dHvA QOs are found beyond the QL of these QOs. Instead, MT linearly increases with respect to H, with a pronounced slope change after entering the QL. The change of the dHvA QOs and the magnetic response at different temperatures for a tilted angle θ = 34.5° are shown in Fig. 2c, d, respectively. The dHvA QOs decay at higher temperatures while τ and MT remain intact beyond the QL. It is noteworthy that the extrapolated intercept of the linear MT is far less than zero, which indicates that this featureless MT in strong magnetic fields is not the same as the commonly observed Lamor diamagnetization proportional to low fields. Consistent with the band structure calculations, our data indicate that the extremal cross sections of the Fermi pockets enclosing the W1 and W2 nodes are 9.5 T and 3.3 T, much smaller than that of NbAs38. The lower Fermi level in TaAs makes the Weyl physics dominate (Supplementary Note 3), namely the chirality and the linearity of the bands are well defined. The well-defined Weyl bands are essential in our calculations for the non-saturating magnetization in the QL. We think that both the W1 and W2 pockets contribute to the observed non-saturating behavior in the QL (Supplementary Note 2).

Fig. 2
figure 2

τ and MT versus magnetic field at different temperatures and angles. a, b τ and MT at 1.8 K at different tilted angles, respectively. Inset in a shows the rotation setup where the angle θ is defined as the tilt off the c-axis. The curves in b have been smoothed. c, d τ and MT in a fixed angle (θ = 34.5°) at different temperatures, respectively. Inset in c shows a zoom-in in low fields where strong temperature dependent dHvA QOs superpose on a parabolic background. Inset in d shows the MT curve at 1.8 K, where two dashed coloured lines show the low- and high-field slopes. The slope takes an obvious enhancement near the QL

Full size image

Differential susceptibility

To understand the non-saturating MT, we plot the differential susceptibility χT = ∂MT/∂H at different angles and temperatures with respect to the field in Fig. 3a,b, respectively. With increasing field, the χT evolves from QOs to a plateau when the QL is achieved at different angles. The height of the plateaus remains intact with increasing temperatures, which is sheerly different from the damping amplitude of the dHvA QOs in lower fields (Fig. 3b). The plateau and the slope show a good linearity of the transverse magnetization in the QL as shown in Fig. 3b. Figure 3c shows that the heights of the plateaus above the QL at different angles can be well fitted by a relation of sin 2θ for θ < 60°, consistent with the expression of τ = VM × μ0H, which can be reformulated as \(\frac{1}{2}{\mathrm{\Delta }}\chi \mu _0H^2V\,{\mathrm{sin}}\,2\theta \,({\mathrm{\Delta }}\chi = \chi _{||} - \chi _ \bot )\).

Fig. 3
figure 3

Differential effective susceptibility (χT) for TaAs. a χT at different angles. b χT for θ = 34.5° at different temperatures. c The heights of the χT plateaus at 33 T versus angles. Red dashed line shows the fitting with a relation of sin 2θ. The error bars were obtained from the standard deviations of intercepts when fitting to the step behaviors in the quantum limit

Full size image

Simulations based on a Weyl Hamiltonian

Now we show that the magnetic signals of TaAs dovetail our calculations for the 3D Weyl fermions in the strong-field limit. Previous calculations only addressed M|| over the full range of magnetic field for different types of band contacting37,43,44, while here we formulate a more general magnetization theory for M||, MT, and τ in an arbitrary angle for Weyl semimetals in which the band structure is also taken into account. The full version of the theory including a version for multiple Weyl nodes is presented in Supplementary Note 2 and here we give the conclusion in the strong-field limit. The Hamiltonian for a single node of 3D massless Weyl fermions can be formulated as

$${\cal H} = v_ap_x\sigma _x + v_bp_y\sigma _y + v_cp_z\sigma _z,$$
(1)

where va,b,c take into account the anisotropy of the Fermi velocity, px,y,z are the momentum, and σx,y,z are the Pauli matrices. Previous studies mainly focused on the magnetization at low fields (the magnetization will be quenched at high fields where the system enters the QL), which includes the Berry paramagnetism for the Weyl semimetals as pointed out in ref. 38. The competition between the Berry paramagnetism and other magnetic signals (Landau diamagnetism and Lamor diamagnetism) may lead to positive or negative torque signals at low fields. Moreover, the recent theoretical calculations on a lattice model revealed that both the magnitude and sign of the orbital contribution of magnetization depend on the exact value of the Fermi energy in a Weyl/Dirac semimetal45. These theoretical analyses are consistent with our measurements for other members of the TaAs family (Supplementary Note 4). On the other hand, the profile of the magnetization in the QL can be more precisely described, and we find the magnetization of this model in the QL (Supplementary Note 2) as

$$M_{||} \propto - B\,{\mathrm{ln}}\,{\mathrm{\Gamma }}$$
(2)

for the parallel component, and

$$M_T \propto - B\,{\mathrm{ln}}\,{\mathrm{\Gamma }}\frac{{\partial {\mathrm{\varDelta }}}}{{\partial \phi }}$$
(3)

for the transverse component. Here \(\Gamma = 2{\varLambda }\ell _B/v_b\sqrt {2{{\varDelta }}}\), where −Λ is the cutoff energy of the valence band, Δ = (va/vb) cos α cos θ + (vc/vb) sin α sin θ, where \(\alpha = {\mathrm{tan}}^{ - 1}\left( {\frac{{v_c}}{{v_a}}{\mathrm{tan}}\, \theta } \right)\) and \(\ell _B = \sqrt {\hbar /eB}\) is the magnetic length. We can clearly see that the non-saturating (Supplementary Eqs. (34) and (35)) magnetization comes from the lower Weyl cone, which is treated as negative energy band in the QL when the magnetization of the higher Weyl cone is quenched38. We denote that the magnetization that comes from the lower Weyl cone is compatible with the Landau theory with negative-energy LBs. Therefore, we conclude that the behavior of magnetization in the QL does not depend on detailed material parameters, and thus it is a universal spectral fingerprint.

Then we check whether our calculation can well fit the experimental results for TaAs. Figure 4a shows that the measured M|| for H||c on another sample in strong pulsed magnetic field. The data can be well fitted by using Eq. (2) where we assume a fixed Nc (Supplementary Note 6). Our theoretical simulation of the formulas for the 3D Weyl fermions reproduces the dHvA QOs at low fields and a significantly enhanced M|| in the QL. For comparison, the M|| for the 3D non-relativistic electrons with parabolic energy dispersion [see Supplementary Eq. (10)] saturates in the QL. On the other hand, we can well fit MT in Fig. 4b by Eq. (3) (Supplementary Note 6). We emphasize that the fittings with a fixed Nc or EF give a similar trend in strong fields, which is distinct from the saturating magnetization of non-relativistic electrons.

Fig. 4
figure 4

Comparison between experiments and theory for the M|| and MT of TaAs. a M|| (black line) and b MT (black line). The blue dashed lines represent the theoretical results with the constraint of the fixed EF in the full range of magnetic field, while the red dashed lines represent the results with the constraint of the fixed Nc in strong fields

Full size image

Discussion

The linear non-saturating magnetization in the QL is of particular interest because it is beyond Landau’s theory of magnetization for classical electrons46 (see Supplementary Note 2 for a fundamental theoretical approach for the classical theory of magnetization). Such feature has never been observed in semimetals with overlapped parabolic electron and hole bulk bands in strong magnetic fields41,47,48,49,50. Our calculations and experiments demonstrate the unique magnetic response in the QL for the linear energy dispersive of the 0th LB of Weyl fermions. Recently emergent topological materials with relativistic quasiparticles for potential applications in general manifest small Fermi surfaces in which the QL can be accessed in a constant magnetic field. Their relativistic nature of quasiparticles is difficult to identify by spectroscopic and electrical transport techniques. The magnetization may serve as a method for detecting relativistic quasiparticles in emergent materials. Finally, we emphasize that our calculation and explanation are based on a single particle scenario. How our theory of one-particle physics can be applied to many-body effects, such as to excitons55, will be a topic of great interest.

Methods

Sample preparation and characterization

TaAs has 12 pairs of Weyl nodes, which are divided into four pairs of W1 and eight pairs of W2. The EF of the as-grown single-crystalline TaAs is close to the two types of Weyl nodes, which are separated 13 meV in energy space, therefore its Fermi surface consists pairs of Weyl electron pockets (Supplementary Fig. 8)51,52. We prepared the single crystals of TaAs by the standard chemical vapor transfer (CVT)53,54 in this study. The large single crystals we used for the magnetic torque is shown as an inset in Supplementary Fig. 1. Its polished surface shows the (001) plane, which was confirmed by X-ray diffraction measurements. The single crystal for parallel magnetization measurements in the pulsed field is as large as 1 × 1 × 5 mm for acquiring the data with higher resolution.

Measurement of magnetic torque

The magnetic torque measurements were performed using capacitive cantilever in water-cooled magnet with the steady fields up to 33 T in the Chinese High Magnetic Field Laboratory (CHMFL), Hefei. In order to estimate the background signal from the cantilever and the cable, the empty cantilever was calibrated on the same conditions. The details of the calibration and measurements are shown in Supplementary Note 1.