Vanishing quantum oscillations in Dirac semimetal ZrTe₅

Significance

Topological materials exhibit a nontrivial Berry phase, experimental determination of which heavily relies on a straightforward phase analysis of quantum oscillations. We report the observation of a striking spin-zero effect in quantum oscillations of topological materials. The concomitant phase inversion underlines a largely overlooked phase factor in previous oscillation analysis of topological materials. Moreover, our results indicate that the Berry phase in ${\mathrm{Z}\mathrm{r}\mathrm{T}\mathrm{e}}_5$ remains nontrivial in the presence of a magnetic field and support a field-driven line node phase.

Abstract

One of the characteristics of topological materials is their nontrivial Berry phase. Experimental determination of this phase largely relies on a phase analysis of quantum oscillations. We study the angular dependence of the oscillations in a Dirac material ${\mathrm{Z}\mathrm{r}\mathrm{T}\mathrm{e}}_5$ and observe a striking spin-zero effect (i.e., vanishing oscillations accompanied with a phase inversion). This indicates that the Berry phase in ${\mathrm{Z}\mathrm{r}\mathrm{T}\mathrm{e}}_5$ remains nontrivial for arbitrary field direction, in contrast with previous reports. The Zeeman splitting is found to be proportional to the magnetic field based on the condition for the spin-zero effect in a Dirac band. Moreover, it is suggested that the Dirac band in ${\mathrm{Z}\mathrm{r}\mathrm{T}\mathrm{e}}_5$ is likely transformed into a line node other than Weyl points for the field directions at which the spin zero occurs. The results underline a largely overlooked spin factor when determining the Berry phase from quantum oscillations.

Graphene is the first Dirac material. Its most celebrated properties, the linear dispersion and chirality, represent some of the essential characteristics of massless Dirac fermions. The latter gives rise to an additional nontrivial Berry phase of $\pi$ to the electron wave function when it completes a closed orbit. This phase has a direct consequence on the phase of quantum oscillations. In presence of a magnetic field, electrons perform cyclotron motion and form quantized Landau levels. The quantization condition requires an accumulated phase change of $2\pi n$ for the wave function on completing a revolution, where $n$ is an integer (1). The additional Berry phase of $\pi$ naturally leads to a $\pi$ phase shift to the quantum oscillations (2). A phase analysis of oscillations thus becomes a straightforward method for determination of the Berry phase. It has been beautifully shown in graphene and extensively used ever since (3, 4).

Discovery of topological materials (e.g., topological insulators, 3D Dirac semimetals, and Weyl semimetals) has further proven the usefulness of such a phase analysis method, as many of these topological phases feature Dirac/Weyl cones in the bulk or on the surface and the Berry phase is closely linked to their topological properties (5, 6). It has been the most widely used method for determination of the nontrivial Berry phase (7–19). Contrariwise, a shift of the oscillation phase from nontrivial $\pi$ to zero has been considered as an indication of a possible topological phase transition (18–22). Note that theories have predicted a rich set of topological phases that a 3D Dirac semimetal can be turned into when subjected to breaking of certain crystal symmetry or time reversal symmetry (23, 24). It would be very interesting to carry out experimental investigation of these transitions, in which the oscillation phase analysis will continue to be valuable.

In this work, we study the angular dependence of the quantum oscillations in a 3D Dirac material, ${\mathrm{Z}\mathrm{r}\mathrm{T}\mathrm{e}}_5$. A striking effect, namely spin zero, has been observed in topological materials, manifested as vanishing oscillations at certain field directions and a concomitant $\pi$ phase shift. The result shows that a spin zero can happen for a Dirac dispersion and highlights a largely overlooked spin factor in determination of the Berry phase in topological materials. Being a consequence of a destructive interference between two Zeeman split bands, the very existence of the spin zero also strongly favors a line node phase over a Weyl point one for ${\mathrm{Z}\mathrm{r}\mathrm{T}\mathrm{e}}_5$ subject to a magnetic field along directions close to the $a$ or $c$ axes.

Results

Magnetoresistance.

An optical image of a device for both electrical and thermoelectric measurements is shown in Fig. 1B, Inset. The same technique has been used to study the thermoelectric response of nanostructured ${\mathrm{C}\mathrm{d}}_3{\mathrm{A}\mathrm{s}}_2$ (25). The armchair gold strip is a microheater for generating a temperature gradient along the sample. With the structure, both the resistivity tensor and the thermoelectric tensor can be measured. ${\mathrm{Z}\mathrm{r}\mathrm{T}\mathrm{e}}_5$ is an orthorhombic layered material stacked along the $b$ axis, while the trigonal prismatic chains of ${\mathrm{Z}\mathrm{r}\mathrm{T}\mathrm{e}}_3$ are along the $a$ axis, which is usually the longest dimension in a single crystal. In this study, the current/temperature gradient is always along the $a$ axis. The temperature $T$ dependence of the resistivity $\rho$, shown in Fig. 1B, displays the characteristic broad maximum of ${\mathrm{Z}\mathrm{r}\mathrm{T}\mathrm{e}}_5$ at $T=124$ K (26–29).

Fig. 1.

Resistivity of ${\mathrm{Z}\mathrm{r}\mathrm{T}\mathrm{e}}_5$. (A) MR for field in the $bc$ plane at $T=1.5$ K. ${\theta }_{bc}=0°$ denotes the $b$ axis, and ${\theta }_{bc}=90°$ denotes the $c$ axis. Inset is a schematic illustration of the geometry for the angle-dependent measurement. (B) Temperature dependence of resistivity. The anomaly, a peak, appears at $T=124$ K. Inset is an optical image of a device that we used for both electric and thermoelectric measurements. (C) MR for field in the $ba$ plane at $T=1.7$ K. ${\theta }_{ba}=0°$ denotes the $b$ axis, and ${\theta }_{ba}=90°$ denotes the $a$ axis.

In the following, we focus on the angular dependence of the magnetoresistance (MR), defined as $MR=\frac{\rho \left(B\right)-\rho \left(0\right)}{\rho \left(0\right)}$. Starting from the $b$ axis, the magnetic field is tilted toward the $c$ ($a$) axis by an angle denoted as ${\theta }_{bc}$ (${\theta }_{ba}$). When the field is parallel to the $b$ axis (${\theta }_{bc}{=\theta }_{ba}=0$), MR is over 1,200$\%$ at 14 T. When the field is tilted away from the $b$ axis toward the $a$ or $c$ axis, MR decreases substantially, as reported by others (21, 30, 31). Particularly when $B\Vert a$, MR is nearly zero, as shown in Fig. 1C.

On top of MR, Shubnikov–de Haas oscillations (SdHOs) are well-resolved. To obtain the oscillatory component $\mathrm{\Delta }\rho \left(B\right)$, a smooth background ${\rho }_0\left(B\right)$ has been subtracted. In Fig. 2A, $\mathrm{\Delta }\rho \left(B\right)/{\rho }_0\left(B\right)$ is plotted as a function of $1/B$ when $B$ is in the $bc$ plane. When $B\Vert b$, the onset of SdHOs is as low as 0.4 T, indicating a relatively high mobility. From the angular dependence of the oscillation frequency, the Fermi surface topography can be mapped out (SI Appendix, Fig. S3). The Fermi surface is a cigar-like ellipsoid, with the longest dimension along the $b$ axis. The Fermi wave vectors in three crystal axes are $k_a=0.096$, $k_b=0.761$, and $k_c=0.124$ nm⁻¹. The strong anisotropic shape agrees well with other studies (21, 30, 33). The frequency is small when the direction of the magnetic field is not far away from the $b$ axis. Therefore, we reach the quantum limit at a relatively low field: $\sim 4$ T. Zeeman splitting starts to appear for the Landau level $n=1$. There are additional weak oscillations in the quantum limit, which have been reported before, and the origin is not clear (21, 30).

Fig. 2.

Angular dependence of SdHOs for field in the $bc$ plane at $T=1.5$ K. (A) MR oscillations after subtracting a smooth background ${\rho }_0\left(B\right)$. Oscillations almost disappear at the angle ${\theta }_{bc}=83.8°$. (Inset) FFT amplitude of the fundamental vs. ${\theta }_{bc}$. The dashed line is a guide to eyes. (B) Landau plots for different field angles. An integer Landau index is assigned to the resistance peak, while a half-integer is assigned to the valley. To obtain the error for the magnetic field positions of the peak and valley, we have adopted twice the noise level as the error for the resistance and then find the corresponding error for the field position from the MR curve. Solid lines are linear fits weighted by errors in $1/B$ (32). (C) Intercept $\gamma$ of the Landau plot as a function of ${\theta }_{bc}$. A sudden change from near $\frac{1}{8}$ to near $\frac{5}{8}$ occurs at the angle between $\mathrm{80}°$ and 85.$\mathrm{5}°$. This angle range overlaps the angle at which the spin zero happens.

Vanishing Quantum Oscillations.

The most interesting feature comes from the angular dependence of the oscillation amplitude. The dependence is not monotonic. At ${\theta }_{bc}\approx 83.8°$, it becomes very weak. This evolution can be seen more clearly in Fig. 2A, Inset, where the fundamental amplitude obtained by a fast Fourier transformation (FFT) is plotted against ${\theta }_{bc}$. A sharp minimum at ${\theta }_{bc}\approx 83.8°$ is evident. In Fig. 2B, a Landau plot is performed for SdHOs at different angles. The oscillation maxima are assigned an integer (14, 34). By a linear fit, the intercept $\gamma$ of the Landau-level index $n$ at $1/B=0$ can be obtained and is plotted as a function of ${\theta }_{bc}$ in Fig. 2C. When ${\theta }_{bc}<83.8°$, $\gamma \approx 1/8$, which suggests a nontrivial Berry phase of $\pi$, in agreement with other studies (21, 30). However, it jumps to around 5/8 when ${\theta }_{bc}>83.8°$. This $\pi$ phase change has previously been observed and attributed to a change of the Berry phase (hence, the topology of the energy band) (21, 35). Intriguingly, this critical angle coincides with that at which the oscillation amplitude drops to zero.

Similar angular dependence has also been observed when $B$ is tilted in the $ba$ plane, as illustrated in Fig. 3. The amplitude of the quantum oscillations displays a minimum of near zero at ${\theta }_{ba}\approx 86.5°$. The angle is slightly larger than that when $B$ is in the $bc$ plane. Concurrently, the intercept $\gamma$ undergoes an abrupt change from 1/8 to about 5/8. In contrast, when $B$ is tilted within the $ac$ plane, similar disappearance of SdHOs and abrupt change of $\gamma$ have not been observed, shown in SI Appendix, Fig. S4. In addition, we have measured the field-dependent thermopower for different field directions (SI Appendix, Figs. S5 and S6). As the thermopower is proportional to the derivative of the conductivity with respect to energy according to the Mott relation, its oscillations in field are often stronger than that of the resistivity (36). Indeed, the suppression of the quantum oscillations to zero is more evident there. The critical angles are the same.

Fig. 3.

Angular dependence of SdHOs for field in the $ba$ plane at $T=1.7$ K. (A) MR oscillations after subtracting a smooth background ${\rho }_0\left(B\right)$. Oscillations almost disappear at the angle ${\theta }_{ba}$ = 86.$\mathrm{5}°$. (Inset) FFT amplitude of the fundamental vs. ${\theta }_{ba}$. The red dashed line is a guide to the eye. (B) Landau plots for different field angles. Solid lines are linear fits. An integer Landau index is assigned to the resistance peak, while a half-integer is assigned to the valley. (C) Intercept $\gamma$ of the Landau plot as a function of ${\theta }_{ba}$. A sudden change from near $\frac{1}{8}$ to near $\frac{5}{8}$ occurs at the angle between $85°$ to $\mathrm{88}°$.

SdHOs are a consequence of formation of Landau levels, which result from the cyclotron motion of electron along closed orbits in the momentum space. In layered materials with sufficiently low interlayer coupling, the Fermi surface is a warped cylinder along the interlayer direction (37–39). Thus, when the field is perpendicular to the axis of the cylinder, the orbit will be open, leading to collapse of Landau levels (hence, SdHOs) (1). In ${\mathrm{Z}\mathrm{r}\mathrm{T}\mathrm{e}}_5$, we and others have already experimentally determined the topology of the Fermi surface being an ellipsoid, which excludes any open orbit (21, 30, 33). In fact, SdHOs are present even when $B\Vert a$ or $c$. An alternative way to understand this is to look at the dependence of the Landau-level separation on the effective mass of carriers. The heavier the carriers, the smaller the Landau-level separation is (hence, a smaller oscillation amplitude). An open orbit indicates an infinite mass, which completely suppresses the Landau level. The effective mass in ${\mathrm{Z}\mathrm{r}\mathrm{T}\mathrm{e}}_5$ increases when the field is tilted away from the $b$ axis toward the $ac$ plane, suggesting a monotonic decrease of the amplitude, which is inconsistent with the observed nonmonotonic dependence. More importantly, the effect cannot explain the phase inversion. Therefore, the Fermi surface topology is not the origin of the vanishing amplitude.

Spin-Zero Effect.

To understand the disappearance of SdHOs, we turn to the Lifshitz–Kosevich formula, according to which the fundamental oscillations are expressed as

$$\frac{\mathrm{\Delta }\rho \left(B\right)}{{\rho }_0}\propto R_T{R}_{\mathrm{D}}{R}_{\mathrm{s}}\cos \left(2\pi \left(\frac{B_f}{B}+\gamma \right)\right),$$ [1]

where $R_T$, $R_{\mathrm{D}}$, and $R_{\mathrm{s}}$ stand for the reduction factors due to temperature, scattering, and spin splitting, respectively. $B_f$ is the oscillation frequency and linked to the extreme cross-section area $S_{\mathrm{e}}$ of the Fermi surface by the Onsager relation $B_f=S_{\mathrm{e}}\hslash /2\pi e$ (1). The shape of the Fermi surface can be inferred from the angular dependence of $B_f$ (SI Appendix, Fig. S3). The phase factor $\gamma =1/2-{\varphi }_{\mathrm{B}}/2\pi -\delta$ is obtained by reading the intercept in a Landau plot. Here, ${\varphi }_{\mathrm{B}}$ is the Berry phase, and $\delta =\pm 1/8$ in 3D systems (14). This is the relation used to determine the Berry phase of a system (3, 4). The spin factor $R_{\mathrm{s}}$ originates from lifting of the twofold spin degeneracy. When Landau levels are spin split, one will have two sets of oscillations (i.e., two frequencies). In general, a beating pattern occurs. $R_{\mathrm{s}}$ describes the envelope of the beating pattern, $R_{\mathrm{s}}=\cos \left(\pi E_{\mathrm{s}}/\mathrm{\Delta }E\right)$, where $E_{\mathrm{s}}$ and $\mathrm{\Delta }E$ are the spin-splitting energy and the Landau-level separation, respectively. For normal nonrealistic electrons, the Landau-level separation is independent of energy and proportional to the field, $\mathrm{\Delta }E=\hslash {\omega }_{\mathrm{c}}=\hslash eB/m^{*}$, where $\hslash$ is the reduced Plank constant, ${\omega }_{\mathrm{c}}$ is the cyclotron angular frequency, $e$ is the electron charge, and $m^{*}$ is the effective mass of electrons. When the spin splitting is due to the Zeeman effect, $E_{\mathrm{s}}=g{\mu }_{\mathrm{B}}B=gBe\hslash /2m_{\mathrm{e}}$, where $g$ is the Landé factor, ${\mu }_{\mathrm{B}}$ is the Bohr magnetron, $m_{\mathrm{e}}$ is the electron mass, and $R_{\mathrm{s}}=\cos \left(\frac{\pi }{2}\frac{g{m}^{*}}{m_{\mathrm{e}}}\right)$ is independent of $B$. In other words, beating vanishes, and only a single frequency remains. Particularly, when $g{m}^{*}/m_{\mathrm{e}}$ is an odd number, $R_{\mathrm{s}}=0$ for arbitrary field. Such a disappearance of SdHOs is the so-called spin zero (1).

The stringent requirement for the spin zero implies that this effect is not commonly seen. If $g$ and/or $m^{*}$ are strongly anisotropic, $g{m}^{*}/m_{\mathrm{e}}$ may vary in a range that includes at least one odd number. Therefore, it is possible to observe a spin zero by rotating the field. ${\mathrm{Z}\mathrm{r}\mathrm{T}\mathrm{e}}_5$ has an anisotropic electronic structure manifested by the cigar-like Fermi surface, as we have just shown. $m^{*}$ also differs significantly for different directions (21, 30). Furthermore, the $g$ factor has been found to be large (40–42), which helps $g{m}^{*}/m_{\mathrm{e}}$ to span a larger range. In this regard, it should not be unexpected to see a spin-zero effect at a certain field angle, as observed in our study. Moreover, an important feature of the spin zero is a simultaneous phase inversion. As the field rotates, $g{m}^{*}/m_{\mathrm{e}}$ passes an odd number. $R_{\mathrm{s}}$ passes zero and then changes its sign, which leads to an inversion of the oscillation phase. The observed change of $\gamma$ from 1/8 to a value close to 5/8 around the zero-oscillation amplitude is in excellent agreement with the spin-zero effect. At the spin zero, we have $g{m}^{*}/m_{\mathrm{e}}=\mathrm{1,3,5,7}\dots$. This is can be further narrowed down to $g{m}^{*}/m_{\mathrm{e}}=\mathrm{1,5,9}\dots$ by recognizing a nontrivial Berry phase ($R_{\mathrm{s}}>0$) when $B\Vert b$. Because the $n=2$ level exhibits spin splitting (for instance, around ${\theta }_{bc}=80.1°$ in Fig. 2A), we have $g{m}^{*}/m_{\mathrm{e}}<4$; hence, $g{m}^{*}/m_{\mathrm{e}}=1$ (1). This choice is further supported by noting that larger $g$ values combined with a strong anisotropic $m^{*}$ most likely lead to multiple spin zeroes, which are at odds with experiments. The cyclotron mass $m_{ac}^{*}=0.026m_{\mathrm{e}}$ is calculated from the temperature-dependent damping of the oscillations. Based on the Dirac dispersion and the ellipsoidal Fermi surface obtained from the angular dependence of the oscillation frequency, the band velocity is calculated. Then, the cyclotron mass at the spin-zero angle is determined: $m^{*}\left({\theta }_{bc}=83.8°\right)=0.132m_{\mathrm{e}}$ (43). Therefore, the $g$ factor is $g\left({\theta }_{bc}=83.8°\right)=7.6$. Similarly, $g\left({\theta }_{ba}=86.5°\right)=5.3$. The g factor has been experimentally determined as 15.8–24.3 in the $b$ direction (21, 41, 42), but these values cannot be directly compared with our results obtained in directions close to the $ac$ plane. A first principle calculation of the g factor has been carried out recently for a few topological materials, by which we obtain $g\left({\theta }_{bc}=83.8°\right)=9.6$ and $g\left({\theta }_{ba}=86.5°\right)=0.3$ (44). The agreement for ${\theta }_{bc}=83.8°$ is reasonably good, while it is not for ${\theta }_{ba}=86.5°$.

Eq. 1 represents only the dominant fundamental component, which cannot produce a spin-splitting feature in SdHOs, unless sufficiently strong higher harmonics are included. In our experiment, when the field is not in the vicinity of the spin-zero angles, splitting of oscillations can be barely seen, except for some weak indication at high fields. However, as the field is tilted toward the spin-zero angles, the splitting becomes increasingly apparent, seen in Fig. 4A. This is because higher harmonics emerge from the vanishing fundamental. At the spin-zero angle, the split peaks are equally spaced, proving that the phases of the fundamental oscillations for the spin-up and spin-down Landau levels differ by $\pi$, as expected by the spin-zero effect. Note that the spin factor for harmonics is $R_{\mathrm{s}}=\cos \left(\frac{p\pi }{2}\frac{g{m}^{*}}{m_{\mathrm{e}}}\right)$, where $p$ is the harmonic order (1). At the spin zero, where the fundamental vanishes, the second harmonic is in its maximum, which is experimentally observed, as seen in Fig. 4B.

Fig. 4.

Zeeman splitting in the vicinity of a spin zero. (A) Oscillations as a function of the Landau index $n$ for tilt angles around the spin-zero angle ${\theta }_{bc}=83.8°$. The phase inversion is evident. The black dashed lines are guide to the eyes, showing the evolution of the Zeeman splitting with the tilt angle. At the spin-zero angle 83.$\mathrm{8}°$, split peaks are equally spaced. (B) FFT amplitude of the second harmonic as a function of the tilt angle. The amplitude peaks at the spin-zero angle.

Spin-Zero Effect in a Relativistic Band.

Although the data agree well with the spin-zero effect, there seems to be one caveat left. The above spin factor $R_{\mathrm{s}}=\cos \left(\frac{\pi }{2}\frac{g{m}^{*}}{m_{\mathrm{e}}}\right)$ is obtained for a nonrelativistic parabolic band. For a Dirac cone, Landau levels are not equally spaced in energy and the energy separations are dependent on $\sqrt{B}$, $E_n=\sqrt{2\hslash e{v}_0^{2}nB}$, where $v_0$ is the band velocity (3, 4, 45). In the case of a $B$-linear Zeeman splitting $E_{\mathrm{s}}=g{\mu }_{\mathrm{B}}B$ (21, 23, 24, 42), a sizable splitting will result in an irregular arrangement between spin-up and spin-down Landau levels, as sketched in SI Appendix, Fig. S8. Will the irregular level arrangement prevent a regular interference between oscillations from two spin bands (hence, a spin zero)? To answer the question, we have taken into account a Dirac dispersion $E=\hslash k{v}_0$ and derived the expression for the spin factor (SI Appendix)

$$R_{\mathrm{s}}=\cos \left(\pi \frac{E_{\mathrm{F}}g{\mu }_{\mathrm{B}}}{\hslash e{v}_0^2}\right).$$ [2]

Note that the cyclotron mass of a massless Dirac particle can be written as $m_{\mathrm{c}}=E_{\mathrm{F}}/v_0^2$. Thus, we arrive at $R_{\mathrm{s}}=\cos \left(\frac{\pi }{2}\frac{g{m}_{\mathrm{c}}}{m_{\mathrm{e}}}\right)$, which looks the same as the one for nonrelativistic electrons. Hence, oscillation interference will take place in the same way. The estimation of the $g$ factor that we have just done in the previous section remains valid. Even so, two differences are worth mentioning. First, to have a spin zero in Dirac materials, a large $g$ factor is required, as $m_{\mathrm{c}}$ is usually very small. While $g$ is too small in graphene, it is often quite large in topological materials due to strong spin orbit coupling (7, 8, 18, 21, 40, 41, 46, 47). Second, $m_{\mathrm{c}}$ in Dirac materials is a function of the Fermi level. Therefore, $R_{\mathrm{s}}$ strongly depends on the Fermi level, which is in stark contrast to that in a parabolic band. Consequently, tuning $E_{\mathrm{F}}$ can be used as a knob to generate a spin zero. Because of the phase inversion accompanied with a spin zero, it should be kept in mind that an observation of such a phase inversion does not indicate a change of the Berry phase.

Implications.

Some interesting conclusions can be drawn based on the observation of the spin-zero effect in ${\mathrm{Z}\mathrm{r}\mathrm{T}\mathrm{e}}_5$. First, it suggests that the Berry phase of ${\mathrm{Z}\mathrm{r}\mathrm{T}\mathrm{e}}_5$ stays unchanged as the field is tilted away from the $b$ axis toward the $ac$ plane. In other words, the Fermi surface is topologically nontrivial for the field applied in any direction. Second, it can be inferred that the Zeeman splitting in ${\mathrm{Z}\mathrm{r}\mathrm{T}\mathrm{e}}_5$ is linear in $B$ (i.e., the effective $g$ factor is independent of $B$). Although this seems a commonplace, it is not necessarily the case in the presence of spin orbit coupling (48). For the 3D Dirac electrons in the ultrarelativistic limit of the quantum electrodynamics (QED), the energy of the Zeeman split Landau levels is

$$E_{n,\sigma }\left(k_z\right)=\pm \hslash c\sqrt{\left(2n+1+\sigma \right)eB/m_{\mathrm{c}}+k_z^2},$$ [3]

where the Landau-level index $n=\mathrm{0,1,2}\dots$, $\sigma =\pm 1$ denotes the spin direction, $c$s the speed of light, and $k_z$ is the wave vector along the field (49, 50). When $k_z=0$, $g=\left(E_{n,\uparrow }-E_{n,\downarrow }\right)/{\mu }_{\mathrm{B}}B$ diverges as $1/\sqrt{B}$. This effect has been observed in a Kane fermion system, HgCdTe (50). Since $E_{n,\uparrow }=E_{n+1,\downarrow }$, the spin factor $R_{\mathrm{s}}=1$ (hence, no spin zero for Dirac electrons in the ultrarelativistic QED). On the contrary, the spin zero always occurs for Kane fermions, because $\sigma$ in Eq. 3 is now replaced by $\sigma /2$ (50). It would be intriguing to check the oscillations in HgCdTe. Third, the appearance of the spin zero sheds light on some aspects of the phase into which ${\mathrm{Z}\mathrm{r}\mathrm{T}\mathrm{e}}_5$, as a 3D Dirac material, is driven by a magnetic field. The 3D topological Dirac phase is at the boundary with various topological phases (23, 24). The Dirac cone consists of two overlapping Weyl cones with opposite chirality and is protected by certain crystal symmetries and time reversal symmetry. Breaking of these symmetries gives rise to different topological phases, depending on which of them is broken and how it is broken (23, 24). For ${\mathrm{Z}\mathrm{r}\mathrm{T}\mathrm{e}}_5$, it is proposed that application of a magnetic field can drive the system into either a Weyl semimetal when $B\Vert a$ or a line node semimetal when $B\Vert b$ or $c$ (42). Because a spin zero requires that the areas of the extreme orbit perpendicular to the field for the two spin-split Fermi surfaces are not equal, this immediately rules out the Weyl node scenario: two Weyl nodes are separated in momentum along the field, and the extreme orbits are identical in shape. However, the spin zero is compatible with a line node phase, in which two Fermi surfaces are separated in energy by $g{\mu }_{\mathrm{B}}B$ (42). As a result, the spin-zero effect can provide clear-cut evidence for distinguishing these topological phases.

Conclusions

In conclusion, we observe a spin-zero effect in the Dirac semimetal ${\mathrm{Z}\mathrm{r}\mathrm{T}\mathrm{e}}_5$ when the direction of the field is close to either the $a$ or $c$ axis. The phase inversion accompanied with the effect indicates that the Berry phase remains nontrivial, in contrast to previous reports. Analysis of the spin-zero effect in a Dirac band suggests that the Zeeman splitting is proportional to $B$ when the spin zero happens. The experiment calls for caution with regard to determination of the Berry phase by quantum oscillations.

Materials and Methods

Microplatelets of ${\mathrm{Z}\mathrm{r}\mathrm{T}\mathrm{e}}_5$ were grown by a silicon-assisted chemical vapor transport method using iodine as a transport agent (33, 51, 52). The growth detail can be found elsewhere (53). Platelets of around 100 nm in thickness were selected in the study. High-angle annular dark-field images taken with the aberration-corrected transmission electron microscope (JEOL ARM200F) show great crystalline quality of our platelets (SI Appendix, Fig. S1). A standard e-beam lithography process was used to fabricate Hall bar structures and thermoelectric measurement devices. Transport measurements were carried out using a lock-in method in a helium cryostat and a Physical Property Measurement System by Quantum Design.