Zeeman splitting and dynamical mass generation in Dirac semimetal

Results

Growth and hall-effect measurements

ZrTe5 single crystals were grown by chemical vapour transport with iodine as reported elsewhere41. The bulk ZrTe5 has an orthorhombic layered structure with the lattice parameters of a=0.38 nm, b=1.43 nm,c=1.37 nm and a space group of Cmcm ()7. The ZrTe5 layers stack along the b axis. In the a–c plane, ZrTe3 chains along a axis are connected by Te atoms in the c axis direction. Figure 1a is a typical high-resolution transmission electron microscopy (HRTEM) image taken from an as-grown layered sample, from which the high crystalline quality can be demonstrated. The inset selected area electron diffraction pattern together with the HRTEM image confirms that the layer normal is along the b axis.

Figure 1: Crystal structure and Hall effect measurements of ZrTe5.

(a) An HRTEM image of ZrTe5 with an inset selected area electron diffraction (SAED) pattern, showing the layer normal along the b axis. The white scale bar corresponds to 2 nm. (b) Temperature-dependent resistance under zero magnetic field. An anomalous resistance peak occurs at T∼138 K. (c) Temperature-dependent Hall resistance of ZrTe5. The nonlinear Hall slopes at both low temperature and high temperature demonstrate the multi-carrier transport in ZrTe5. (d) The temperature-dependent mobility and carrier density of the dominant carriers. A transition of electron- to hole-dominated transport is observed around the temperature of the anomalous resistance peak. The graduated background represents the amount and type of carriers, blue for holes and red for electrons.

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The temperature dependence of the resistance Rxx and Hall effect measurements provide information on the electronic states of a material in a succinct way. We first carried out regular transport measurements to extract the fundamental band parameters of as-grown ZrTe5 crystals. In a Hall bar sample, the current was applied along the a axis and the magnetic field was applied along the b axis (the stacking direction of the ZrTe5 layers). Figure 1b shows the temperature dependence of the resistance Rxx of ZrTe5 under the zero field. An anomalous peak, the unambiguous hallmark of ZrTe5 (ref. 4), emerges at around 138 K and is ascribed to the temperature-dependent Fermi energy shift of the electronic band structure42. The Hall effect measurements provide more information on the charge carriers responsible for the transport. The Hall coefficient changes sign around the anomaly temperature, implying the dominant charge carriers changing from holes to electrons (Fig. 1c). The nonlinear Hall signal suggests a multi-carrier transport at both low and high temperatures, which is also confirmed by the Kohler’s plot and our first-principles electronic structure calculations (Supplementary Note 1 andSupplementary Figs 1–5). For convenience, a two-carrier transport model43,44 is adopted to estimate the carrier density and mobility. The dominant electron exhibits an ultrahigh mobility of around 50,000 cm2 V−1 s−1 at low temperature, which leads to strong SdH oscillations as we will discuss later. Around the temperature of the resistance anomaly, the electron carrier density has already decayed to almost one-tenth of that at low temperature, and finally holes become the majority carriers at T>138 K (Fig. 1d). Detailed analysis of the two-carrier transport is described in Supplementary Notes 1 and 2 and in Supplementary Figs 1–8.

Fermi surface and quantum oscillations analysis

Elaborate measurements of angle-dependent magnetoresistance (MR) provide further insight into the band-topological properties of ZrTe5. A different external magnetic field geometry has been exploited to detect the Fermi surface at 2 K, as shown in Fig. 2. When the magnetic fieldB>0.5T is applied along the b axis, clear quantum oscillations can be identified, indicating a high mobility exceeding 20,000 cm2 V−1 s−1. The MR ratio  is around 10 (here R(B) is the resistance under magnetic field B and R(0) is the resistance under zero field), lower than previous results8 on account of different Fermi level positions. As the magnetic field is tilted away from the b axis, the MR damps with the law of cosines, suggesting a quasi-2D nature and a highly anisotropic Fermi surface with the cigar/ellipsoid shape. This is reasonable for a layered material45. A Landau fan diagram of arbitrary angle (Fig. 2b) is plotted to extract the oscillation frequency SF and Berry’s phase ΦB according to the Lifshitz–Onsager quantization rule46: , where N is the Landau level index, SF is obtained from the slope of Landau fan diagram and γ is the intercept. For Dirac fermions, a value of |γ| between 0 and 1/8 implies a non-trivial π Berry’s phase46, whereas a value of around 0.5 represents a trivial Berry’s phase. Here the integer indices denote the ΔRxx peak positions in 1/B, while half integer indices represent the ΔRxx valley positions. To avoid the influence from the Zeeman effect, here we only consider the N≥3 Landau levels. With the magnetic field along the b axis, the Landau fan diagram yields an intercept γ of 0.14±0.05, exhibiting a non-trivial Berry’s phase for the detected Fermi surface. At the same time, SF shows a small value of 4.8 T, corresponding to a tiny Fermi area of 4.6 × 10−4 Å−2. The system remains in the non-trivial Berry’s phase as long as 0≤β≤70° (Fig. 2b inset and Fig. 2c). We have also obtained the angular dependence of SF as illustrated in the inset of Fig. 2c where a good agreement with a 1/cosβrelationship is reached, confirming a quasi-2D Fermi surface. However, as the magnetic field is rotated towards the c axis (β>70°), the Berry’s phase begins to deviate from the non-trivial and finally turns to be trivial when B is along the c axis (Fig. 2b inset, the two dark-red curves with intercept of ∼0.5). Meanwhile, the oscillation frequency SF deviates from the cosines law and gives a value of 29.4 T along the c axis (Fig. 2cinset). The Berry’s phase development along with the angular-dependent SF unveils the quasi-2D Dirac nature of ZrTe5 and possibly a nonlinear energy dispersion along the c axis. This is also confirmed by the band parameters such as the effective mass and the Fermi velocity, as described below.

Figure 2: Angular MR and SdH oscillations of ZrTe5.

(a) Angular MR of ZrTe5. The inset shows the geometry of external magnetic field. (b) Landau fan diagram of arbitrary angle in a. Inset: Zoom-in view of the intercept on y axis. (c) The angular-dependent intercept of Landau fan diagram in b. Inset: angular-dependent oscillation frequency. The error bars were generated from the linear fitting process in the Landau fan diagrams. (d,e) The effective mass of ZrTe5 when the magnetic field is applied along b axis and c axis, respectively. The error bars were generated from the fitting process. (f) The quantum oscillations of Rxx and quantized plateaus in Rxy.

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A meticulous analysis of the oscillation amplitude at different angles was conducted to reveal the electronic band structure of ZrTe5. Following the Lifshitz–Kosevich formula46,47,48, the oscillation component ΔRxx could be described by

where RT, RD and RS are three reduction factors accounting for the phase smearing effect of temperature, scattering and spin splitting, respectively. Temperature-dependent oscillation ΔRxx could be captured by the temperature smearing factor , wherekB is the Boltzmann’s constant, ħ is the reduced Plank’s constant and m*is the in-plane average cyclotron effective mass. By performing the best fit of the thermal damping oscillation to the equation, the effective mass ma–c* (when the magnetic field is applied along the b axis, the Fermi surface in a–c plane is detected) is extracted to be 0.026me, where me is the free electron mass (Fig. 2d). Such a small effective mass agrees well with the Dirac nature along this direction; and it is comparable to previously reported Dirac18,49,50 or Weyl semimetals19. The corresponding Fermi velocity yields a value of 5.2 × 105 m s−1, which agrees with recent ARPES results42. A similar analysis gives a value ofma–b*=0.16me and mb–c*=0.26me, respectively (Fig. 2e; also seeSupplementary Note 3, Supplementary Figs 9–11 and Supplementary Table 1 for detailed information). Both ma–b* and mb–c* are larger thanma–c*; this indicates a deviation from linear dispersion of these two surfaces considering the weak interlayer coupling7, which is in agreement with our previous results51 and the reported ARPES8. The carrier lifetime τ could be obtained from the Dingle factor RD∼e−D, where . Table 1 summarizes the analysed parameters of the band structure.

Table 1: Band parameters of ZrTe5.

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Besides the obvious SdH oscillations of Rxx, Rxy exhibits distinct nearly quantized plateaus, whose positions show a good alignment with the valley of Rxx (Fig. 2f). The value of 1/Rxy establishes a strict linearity of the index plot and demonstrates the excellent quantization (Fig. 2finset), reminiscent of the bulk quantum Hall effect. A similar behaviour has been observed in several highly anisotropic layered materials, such as the heavily n-doped Bi2Se3 (ref. 52), η-Mo4O11 (ref. 53) and organic Bechgaard salt54,55,56. At variance with the quantum Hall effect in a 2D electron gas, the quantization of the inverse Hall resistance does not strictly correspond to the quantum conductance. In fact, because of the weak interlayer interaction7, bulk ZrTe5 behaves as a series of stacking parallel 2D electron channels with layered transport, which leads to the 2D-like magneto-transport as discussed above. The impurity or the coupling between the adjacent layers in the bulk causes the dissipation so that the Rxx cannot reach zero52. It is worth noting that the peak associated to the second Landau level in Rxx displays a broad feature with two small corners marked by the arrows, implying the emergence of spin splitting.

Zeeman splitting under extremely low temperature

It is quite remarkable that the spin degeneracy can be removed by such a weak magnetic field. To investigate the spin splitting, it is necessary to further reduce the system temperature. Figure 3a shows the MR behaviour of ZrTe5 at 260 mK. A peak deriving from the first Landau level can be observed at ∼6 T. The second Landau level offers a better view of the Zeeman splitting because of the relatively small MR background, as marked by the dashed lines in Fig. 3b. The Rxy signal provides a much clearer signal: after subtracting the MR background, it reveals strong Zeeman splitting from the oscillatory component ΔRxy(Fig. 3c and Supplementary Fig. 12). At 0.4 K, the fifth Landau level begins to exhibit a doublet structure with a broad feature. Under higher magnetic fields, the separation of the doublet structure increases, in particular, the second Landau level completely evolves into two peaks, indicating the complete lifting of spin degeneracy due to the Zeeman effect. To analyse the Zeeman effect occurred at such low temperature conveniently, we rearranged the spin phase factor in equation (1) by the product-to-sum formula47 (detailed mathematical process is available in Supplementary Note 4). As a result, the oscillation component ΔRxx is equivalent to the superposition of the oscillations from the spin-up and spin-down Fermi surface

Figure 3: Zeeman splitting in ZrTe5.

(a) MR behaviour of ZrTe5 at the temperature range of 0.26–3 K. (b) Temperature-dependent MR of ZrTe5. Two dashed lines are a guide to the eyes, which indicate the Zeeman splitting of the second Landau level. (c) The oscillation component in Rxy at 0.4 and 0.5 K. Sizable Zeeman splitting can be distinguished from the second and third Landau levels. (d) Landau fan diagram for both spin-up and spin-down electrons. (e) Angular dependence of the first-order differential Rxy versus 1/Bcosα. (f) The spacing of Zeeman splitting in the second Landau level at different field angles. The inset shows the geometry of external magnetic field.

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where  is the phase difference between the oscillations of spin-up and spin-down electrons. With this method we can estimate the gfactor by Landau index plot for both spin ladders (Fig. 3d). This leads to the g factor of 21.3, in good agreement with the optical results9. Given such a large g factor, it is understandable that the Zeeman splitting could be easily observed in a relatively weak magnetic field. We have further carried out the theoretical Landau level calculations, which provides a clear insight into the Zeeman splitting as elaborated inSupplementary Note 5. In short, when the magnetic field is along the baxis (z direction), the Landau level energy eigenvalues for n≠0 are , where μB is the Bohr magneton, and in this case Ek=ħvzkz,  is the Landau level energy of the band bottom of the n=1 Landau level. Here the Landau levels are split by, resulting in the observed Zeeman splitting.

The angular-dependence of the Zeeman splitting can provide valuable information to probe the underlying splitting mechanism. Figure 3eshows the first-order differential Rxy as a function of 1/Bcosα. Pronounced quantum oscillations with Zeeman splitting can be unambiguously distinguished and they align well with the scale of 1/Bcosα, further verifying the quasi-2D Fermi surface as mentioned before. It is noticeable that the spacing of the Zeeman splitting changes with the field angle. Generally, Zeeman splitting effect is believed to scale with the total external magnetic field so that the spacing of the splitting Landau level would not change with angle. However, in the case of ZrTe5, the spacing of the Zeeman splitting, normalized by Bcosα, is consistent with the quasi-2D nature (Fig. 3f). The angular dependent Zeeman splitting can be attributed to the orbital contribution caused by strong spin-orbit coupling in ZrTe5 (ref. 7). Regarding the effect of the exchange interaction induced by an external field, we may decompose the splitting into an orbital-dependent and an orbital-independent part, where the former one depends on the shape of the band structure that leads to the angle-dependent splitting, and the latter one mainly comes from the Zeeman term that hardly contributes to any angular dependent splitting11. Owing to the highly anisotropic Fermi surface of ZrTe5 and the strong spin orbital coupling from the heavy Zr and Te atoms, the orbital effect in the a–c plane of ZrTe5 is significant, giving rise to a highly anisotropic g factor and an angular-dependent Zeeman splitting. Similar phenomena have also been observed in materials such as Cd3As2 (refs 49, 57) and Bi2Te3 (ref. 58).

Magnetotransport under high magnetic fields

A high magnetic field up to 60 T was applied to drive the sample to the ultra-quantum limit to search for possible phase transitions. Figure 4ashows the angular-dependent MR of ZrTe5 under strong magnetic field. The measurement geometry can be found in Fig. 3f inset. Several features are immediately prominent. First, when the magnetic field is along the a axis, Zeeman splitting is observed, which is consistent with the theoretically solved Landau levels (Supplementary Note 5). The gfactor along the a axis extracted by formula (2) is 3.19, which agrees well with the anisotropy of g factor discussed above (Supplementary Fig. 13). When the magnetic field is along the c axis, spin-splitting is hardly observed, again consistent with the theoretical expectations and the recent magneto-spectroscopy results9 (Supplementary Note 6 andSupplementary Fig. 14). Here the Landau level energy eigenvalues become , here Ek=ħvyky. The effect of the magnetic field is the horizontal shifting of the degenerate Landau levels by ±gμBB/ħvy in the ky vector direction (the field direction), instead of splitting the heights of the band bottom like the case when the field is along the a or b axis, thus there is no Landau level splitting in the quantum oscillation. Second, even with an external magnetic field up to 60 T (in the quantum limit regime), the MR feature still follows theBcosα fitting, confirming once again the quasi-2D nature of ZrTe5 (Fig. 4b). Finally, and most importantly, a huge resistance peak emerges at around 8 T, followed by a flat valley between 12 and 22 T, then the resistance increases and forms a shoulder-like peak at ∼30 T. It should be emphasized that the amplitude of the resistance at 8 T is much larger than the amplitude of SdH oscillations, so that the signal of the first Landau level has been submerged into the anomalous peak. Only a few materials show analogous field-induced electronic instabilities, such as bismuth23,24,26, graphite25,29,59, and more recently the Weyl semimetal TaAs (ref. 60). As we have remarked, the SdH peak due to then=1 Landau level merges into the anomalously large peak around B=8 T, which suggests that the n=2 Landau levels are empty and the electrons in the n=1 Landau level are responsible for the anomalous peak. The location of Fermi level is schematically shown in Fig. 4c. The peak can be naturally explained by the picture of dynamical mass generation (accompanied by a density wave formation, with the wave vector being the nesting vector) in the n=1 Landau levels, which leads to the generation of an energy gap for the electrons in these Landau levels, thus significantly enhancing the resistivity. In Fig. 4c, we illustrate one of the possibilities of nesting vectors responsible for this instability (the other possibility being two vectors connecting the Fermi momenta 1 to 3, and 2 to 4, respectively). Since the vector  is slightly different from the vector  due to the Zeeman splitting, the density wave transition should also be Zeeman split, which is presumably responsible for the existence of a ‘bump’ near the top of the peak in Fig. 2a.

Figure 4: Ultra-quantum limit transport and dynamical mass generation of ZrTe5 at 4.2 K.

(a) Angular-dependent MR of ZrTe5 at 4.2 K under high magnetic field up to 60 T. (b) Angular-dependent MR as a function of effective magnetic field perpendicular to a–c plane. (c,d) The Landau levels and Fermi levels for B≈9 and 25 T, respectively. The inset of d is an illustration of the spin density wave from the n=0 Landau level.