Abstract
The nature of Fermi surface defines the physical properties of conductors and many physical phenomena can be traced to its shape. Although the recent discovery of a currentdependent nonlinear magnetoresistance in spinpolarized nonmagnetic materials has attracted considerable attention in spintronics, correlations between this phenomenon and the underlying fermiology remain unexplored. Here, we report the observation of nonlinear magnetoresistance at room temperature in a semimetal WTe_{2}, with an interesting temperaturedriven inversion. Theoretical calculations reproduce the nonlinear transport measurements and allow us to attribute the inversion to temperatureinduced changes in Fermi surface convexity. We also report a large anisotropy of nonlinear magnetoresistance in WTe_{2}, due to its low symmetry of Fermi surfaces. The good agreement between experiments and theoretical modeling reveals the critical role of Fermi surface topology and convexity on the nonlinear magnetoresponse. These results lay a new path to explore ramifications of distinct fermiology for nonlinear transport in condensedmatter.
Introduction
Layered transition metal dichalcogenides are an emerging class of materials with novel physical phenomena and a wide range of potential applications^{1,2}. Among them, semimetal tungsten ditelluride (WTe_{2}) especially showed an extremely large and unsaturated magnetoresistance (MR)^{3}, which was attributed to a Fermi surface with perfectly compensated electron and hole pockets^{3,4,5}. Up to now, the MR investigated in WTe_{2} appears in the linear region and the exploration of a currentdependent nonlinear magnetoresistance is lacking. In this context, a nonlinear magnetoresistance (NLMR) that scales linearly with the applied electric and magnetic fields has been recently discovered independently in a polar semiconductor^{6} and a topological insulator^{7} with spinmomentum locked bands, and has led to a surge of interest within condensedmatter physics toward understanding its underlying mechanisms^{8,9,10}. In WTe_{2}, the large spinorbit coupling (SOC) and broken inversion symmetry lift the spin degeneracy, as observed in spin and angleresolved photoemission spectroscopy (ARPES)^{11,12,13}. Taking advantage of the large SOC and low crystalline symmetry, WTe_{2} was recently demonstrated as an intriguing spinsource for generating outofplane antidamping torques to an adjacent magnetic material^{14}. To further explore ramifications of the spinpolarized bands in WTe_{2}, and their interplay with the richly structured Fermi surface^{15,16,17}, we investigate in detail the spindependent nonlinear magnetotransport.
WTe_{2} exhibits properties remarkably sensitive to temperature as a combined result of nearly perfect carrier compensation at low temperatures with a holesuppressing Lifshitz transition at ~ 160 K^{18}. Accordingly, small temperature variations cause essential changes in the Fermi surface, as revealed by ARPES^{5,18}. In addition, WTe_{2} undergoes a topological transition from a Weyl to a topologically trivial semimetal at ~ 70 K^{17,19}. While an understanding of the Fermi surface is critical to describe and explore many technologically promising physical phenomena, such as the oscillatory exchange coupling or superconductivity^{20}, and even though previous studies investigated the role of spin textures in both k and real spaces^{6,7,9,10}, the fermiological implications and opportunities remain unexplored in the context of nonlinear magnetotransport. With such a rich and easily tunable Fermi surface, WTe_{2} is an excellent platform to investigate these questions.
In this work, we report the observation of a currentdependent NLMR in WTe_{2} that scales with the first power of both the applied electric current and magnetic field^{6,7}. This is in striking contrast with the conventional (linear) MR so far characterized in WTe_{2}, which is currentindependent and quadratic in the magnetic field^{3,4}. Interestingly, the NLMR shows a temperaturedriven inversion and a significant anisotropy along different crystallographic axes. Our experimental results are reproduced qualitatively by theoretical modeling that combines abinitio band structure calculations with a semiclassical calculation of the magnetoresponse. The calculations reveal that the spinpolarized electronic structure evolves with the magnetic field, giving rise to the measured spindependent NLMR. Furthermore, we establish that the inversion of NLMR arises from a transition in the Fermi contours from convex to concave, whereas the giant anisotropy is due to the low symmetry of the Fermi surface. Therefore, we establish here a close relationship between Fermi surface topology, convexity and the nonlinear magnetotransport response. These results also demonstrate that fine tuning of the Fermi level is critical in controlling the nonlinear magnetotransport in semimetals.
Results
Sample and characterization
At ambient conditions, bulk WTe_{2} is in the orthorhombic T_{d} phase^{21} where, as a result of strong octahedral distortions and the displacement of the metal within each octahedron, the W ions organize as effective zigzag chains along the a axis and sandwiched between Te layers as shown in Fig. 1a. The band structure calculated from the density functional theory (DFT) is plotted in Fig. 1b, and agrees with previous calculations^{18,22}. In Fig. 1c, we show a schematic representation of the Fermi surface with only electron pockets at different carrier filling that illustrates how the sign of the NLMR depends on whether the Fermi surface consists of a concave set or a convex set. In bulk WTe_{2}, the Fermi surface at 0 K consists of multiple hole and electron pockets, which can be seen below.
Our WTe_{2} flakes were obtained by mechanical exfoliation on Si/SiO_{2} substrates and subsequently patterned into Hall devices for transport measurements (see Methods). Multiple devices were prepared with the current channels (that define our x direction) designed along different crystallographic directions of the underlying WTe_{2}. The typical temperature dependence of their resistivity (ρ) is shown in Fig. 1d, and is consistent with previous reports^{3,13}. To first order in the applied current I, the longitudinal resistance R can be expressed as R = R_{0 }+ R′I, where R_{0} is the currentindependent (linear) resistance and R′I is the currentdependent (nonlinear) resistance. In order to study the nonlinear magnetotransport, a lowfrequency a.c. current I_{ω} = I sin(ωt) was applied through the devices (Fig. 1e, f), and we recorded the secondharmonic longitudinal voltage \(V_{2\omega }{\mathrm{ = }}\frac{1}{2}R{\prime}I^2\sin (2\omega t  \pi /2)\) using lockin techniques. The secondharmonic resistance R_{2ω} (\(R_{2\omega } \equiv \frac{1}{2}R{\prime}I\)), equivalent to half of the nonlinear resistance R′I, is used to quantify the nonlinear magnetotransport response (which we verified to be independent of the driving frequency)^{23}. The measurements were performed while rotating the applied magnetic field H in the xy plane of the film (outofplane misalignment evaluated to be <1°) at an angle φ with the current direction, as illustrated in Fig. 1f. We note that, unlike the colossal linear MR, which is the strongest under perpendicular fields^{3,4,5}, the nonlinear magnetoresponse is maximized under inplane fields due to the planar spin texture of bulk WTe_{2}, as we discuss below.
Nonlinear magnetoresistance in WTe_{2}
We observe a sinusoidal dependence of the secondharmonic resistance R_{2ω} on the angle φ between the magnetic field and the current direction. Figure 2a shows that the period is 360° with a maximum when the field is orthogonal to the current (φ = 90° or 270°), and approaches zero when they are collinear (φ = 0° or 180°). Hence, unlike the first harmonic resistance R_{ω} (see Supplementary Fig. 1), R_{2ω} changes sign upon reversal of the magnetic field. To further characterize the NLMR, R_{2ω} was measured at different I and H at 300 K, and the same dependence R_{2ω} = −ΔR_{2ω} sin φ was observed. Figure 2b shows that the extracted amplitude ΔR_{2ω} increases linearly with I and, at the same time, scales linearly with H, as plotted in Fig. 2c (the behavior shown in Fig. 2a–c persists down to 2 K). This contrasts with the field dependence of the linear resistance R_{ω}, which grows quadratically with H (Supplementary Fig. 1)^{3}. These observations extend the class of systems bearing currentdependent nonlinear magnetotransport to the layered semimetal WTe_{2}, where it is observable at room temperature.
Temperaturedriven inversion of the nonlinear magnetoresistance
Motivated by the strong temperature dependence of its other known transport properties^{5,17,18,19}, we study in detail the NLMR of WTe_{2}. Measurements of the φdependent R_{2ω} at different temperatures (Supplementary Fig. 2) reveal that, upon lowering T from room temperature, the initially positive amplitude ΔR_{2ω} undergoes a gradual reduction until it reaches zero at T ≈ 140 K, at which point it changes sign and progressively increases to large negative values as the temperature reduces. This behavior is shown in Fig. 2d, which is distinct from that of the linear MR (see Supplementary Fig. 1). We note that the inversion in R_{2ω} occurs at a temperature close to that of the reported Lifshitz transition in bulk WTe_{2}^{18}, where the Fermi surface topology changes. In order to evaluate changes of the electronic structure with temperature^{18,19}, we measure the Hall resistance R_{xy} in Fig. 2e, which displays a gradual deviation from the linear field dependence at low temperatures. By fitting R_{xy} (H) according to a twocarrier model^{19}, we extract the hole (electron) density, which increases (decreases) upon lowering the temperature (Fig. 2f). This is consistent with the strong sensitivity of the chemical potential to temperature changes in WTe_{2}^{5,18}. As we show below, the temperaturedriven changes in the chemical potential are reflected not only in variations of the size of the electron and hole pockets^{5,18}, as schematically shown in the inset of Fig. 2d, but also in the convexity of the Fermi contours. The latter effect turns out to be crucial to drive the inversion of the NLMR.
Giant crystal anisotropy of nonlinear magnetoresistance
The strong local distortion of the W ions in the T_{d} structure causes them to arrange along onedimensional zigzag chains parallel to the a axis within each monolayer, and this imparts a strong electronic anisotropy^{3,5}. To investigate its consequences in the NLMR R_{2ω}, we pattern circular Hall devices as shown in Fig. 3a, in which currents can be applied along different crystallographic directions of the same WTe_{2} device. One of the current channels is chosen parallel to the a axis of WTe_{2} (Fig. 1a), as identified by polarized Raman spectra^{24} (see Supplementary Fig. 3). The longitudinal (linear) resistivity ρ is plotted in Fig. 3b and, as expected, is anisotropic with values along the a axis ~ 3 times lower than those along b^{25}. The behavior of the secondharmonic R_{2ω} is, however, more interesting as plotted in Fig. 3c for the four different directions of current flow. The magnitude of R_{2ω} is much stronger along b compared to that of the a direction, with a b/a magnitude ratio ~18 at 2 K and ~5 at 300 K. Moreover, the signchange in R_{2ω} is also sensitive to the current direction in relation to the crystallographic axes: the sign inversion with temperature is clear when the current flows along the b axis, and inversion is absent when it flows parallel to a. We define χ = 2 R_{2ω}/(R_{ω} I H), which characterizes the NLMR under unit electric voltage and magnetic field shown in Fig. 3d, and note that χ reaches up to 0.04 and −0.22 Ω V^{−1} T^{−1} (or A^{−1} T^{−1}) along the a and b directions at 2 K, respectively. Such a large anisotropy in the NLMR constitutes a record, and has not been observed in other materials^{6,7,9,10}. The discovery of this giant anisotropy from a nonlinear magnetotransport perspective enriches our understanding of the anisotropic Fermi surface in WTe_{2} as we discuss below.
Theoretical modeling
To gain insight into the observed NLMR in WTe_{2}, we calculated the longitudinal secondorder current density, \(J_{xx}^{(2)}\), in the presence of an external magnetic field H perpendicular to E (Supplementary Note 1), and use the fact that \(J_{xx}^{(2)} \propto  R_{2\omega }\) (see Supplementary Note 2). The calculation relies on a Wannier tightbinding Hamiltonian that reproduces all the details of the DFT band structure of bulk WTe_{2} given in Fig. 1b, including the relative positions of hole and electron pockets, as well as the flat bands immediately below the Fermi energy that were discussed in ref. ^{5}. It is important to note that, in semimetallic WTe_{2}, the Fermi level µ is known to change by ~ 50 meV between 40 and 160 K^{18}. To corroborate this, we have calculated the hole (n_{h}) and electron (n_{e}) densities at different µ (Supplementary Note 3), which are shown with the experimental ones in Fig. 2f. We establish the relation between T (bottom axis in Fig. 2f) and µ (top axis in Fig. 2f) by comparing the calculated n_{e} and n_{h} with the experimental data.
Figure 4a displays the dependence of calculated J^{(2)} on µ at different temperatures for current flowing along the b direction. The sign of \(J_{bb}^{(2)}\) changes from positive to negative with increasing µ, and the inversion threshold is mostly insensitive to the thermal broadening. This indicates that the experimentally observed inversion in R_{2ω} is primarily due to changes in the Fermi level. For a direct comparison, we plotted in Fig. 4d the calculated ratio \(J_{ii}^{(2)}/[J_{ii}^{(1)}]^2 \propto  \chi\) (Supplementary Note 2) versus T for the current along the a and b axes (i = a, b). The calculated result captures the overall qualitative behavior of the experimental data in Fig. 3d, with the sign inversion and large crystalline anisotropy. In addition, the theoretical J^{(2)} in Fig. 4e reproduces the experimental sinusoidal dependence on φ (Fig. 2a), and the linearity of J^{(2)} in Fig. 4f with respect to the magnetic field (Fig. 2c). Another significant implication from the calculations is that the sign inversion of \(J_{bb}^{(2)}\) is dominated by the electron carriers as can be seen from the carrierresolved \(J_{bb}^{(2)}\) in Supplementary Fig. 4. This indicates that the electron pockets are responsible for the signchange.
We now discuss the role of the Fermi surface topology to understand the nontrivial transport phenomena, NLMR in WTe_{2}. The calculated Fermi contours are plotted in Fig. 4b, c at two Fermi levels associated with opposite J^{(2)} in Fig. 4a. In addition to the suppression of hole pockets, we find a distinctive change in the threedimensional (3D) convexity of the Fermi surface: the Fermi contours contain portions with a concave (CC) shape at µ = 0 (Fig. 4b), which evolves into entirely convex (CV) contours at µ = 120 meV (Fig. 4c). As J^{(2)} is governed by both the local band velocity and curvature, the Fermi surface convexity should determine the sign of the nonlinear current. To further illustrate this characteristic observed consistently both in the experiments and the calculations, we build a simplified quasibulk model Hamiltonian^{26} based on symmetry (see Supplementary Fig. 7 and Supplementary Note 4). Reflecting what is seen in the DFTderived band structure (Fig. 4b, c), this model yields electron pockets whose Fermi surface convexity changes with the chemical potential. We calculate \(J_{bb}^{(2)}\) based on this simpler, strippeddown Hamiltonian and see the same sign inversion at a threshold µ, as is clear from Fig. 4g. Moreover, the inversion threshold is directly correlated to the transition of the Fermi contour from CV to CC, as illustrated in Fig. 4h, i, whereas at low µ (Fig. 4i) the longitudinal dispersion is parabolic, at higher µ (Fig. 4h) it becomes shaped like a Mexican hat, a direct manifestation of the CC of the Fermi contour. Thus, it can be concluded that a transition from CV to CC is a necessary condition for the observation of a sign inversion in J^{(2)}, as intuitively expected from the analytical expression for \(J_{bb}^{(2)}\), which depends on the effective mass only along the b direction (see Supplementary Note 5).
This establishes the convexity of the Fermi surface as the driving mechanism behind the sign inversion in J^{(2)} with temperature and indicates that the NLMR is a simple transport observable to electrically monitor the variation of the Fermi level with temperature in this system. Overall, our results confirm the critical role of topology and convexity of the Fermi surface on the exotic nonlinear magnetotransport of WTe_{2}.
Discussion
In semimetallic WTe_{2}, we have demonstrated that it exhibits a NLMR at room temperature, which is sensitive to temperature including a sign reversal, and strongly anisotropic. These properties critically depend on the Fermi surface morphology. In WTe_{2}, the strong temperature dependence arises from an unusually large thermal shift of the chemical potential.
Nonlinear magnetocurrents associated with strong SOC and Fermi surfaces with nontrivial spin textures are a nascent field of research. Here, we have shown the ability to control their magnitude, and especially their sign, with either magnetic fields, or temperature in WTe_{2}. In addition, doping^{22}, pressure^{27}, electrostatic gating^{15,28,29}, and film thickness^{30,31,32} are known to be effective to tune the Fermi surface of WTe_{2}. Reducing the thickness was found to modify the electronic structure of WTe_{2} films^{32}, and flakes with different thicknesses display different R_{2ω} inversion temperatures (Supplementary Fig. 9). The extreme case of monolayer 1 T’WTe_{2} was recently shown to be a twodimensional topological insulator with an insulating bulk and a topologically nontrivial metallic edge^{26,33,34,35}, although the nonlinear magnetotransport at the monolayer edge is still an open question. On the other hand, bulk WTe_{2} was predicted to be a typeII Weyl semimetal^{36}, but the nonlinear magnetoresponse when the Fermi energy of a Weyl semimetal is tuned to the Weyl nodes remains unknown.
All these factors suggest ample space for further tunability and possibly even richer nonlinear, spindependent features in the charge and spin transport of WTe_{2}. Moreover, changes in the topology and convexity of the Fermi surface are widely observed in other materials, such as the Dirac semimetal ZrTe_{5}^{37} and LaAlO_{3}/SrTiO_{3}^{38} at the Lifshitz transition. Details of the Fermi surface are also very sensitive to the strength of SOC in Weyl semimetals of the transition metal monopnictide family^{39}, such as TaAs^{40}, TaP^{41}, and NbP^{39}. We anticipate the existence of a NLMR in these materials as well which, if confirmed, could establish this effect as a transportonly probe of topological and other types of Fermi surface transitions.
Methods
Sample preparation
WTe_{2} thin film flakes with different thicknesses were cleaved from a bulk WTe_{2} single crystal (HQ Graphene) by mechanical exfoliation onto a 300 nm Si/SiO_{2} substrate with alignment markers. Subsequently, a capping layer of SiO_{2} (4 nm) was deposited on them as a protection layer to eliminate material degradation during subsequent fabrication processes. Hall bar devices with a conventional rectangular (as well as radial multiterminal) geometry were fabricated using the standard photolithography (Ebeam lithography) method and Ar^{+}milling. Prior to the Cr (30 nm)/Au (60 nm) electrode deposition using magnetronsputtering, the contact regions were treated with Ar^{+}etching to remove the SiO_{2}. Ohmic contacts were confirmed by I–V measurements. The crystallographic direction of the current channel was confirmed by performing polarized Raman measurements with a misalignment < ± 5°. The flake thickness and surface morphology were measured by an atomic force microscope.
Electrical measurements
Hall bar devices were wirebonded to a rotatable sample holder and installed in a physical property measurement system (PPMS, Quantum Design) for transport measurements in the temperature range 2–300 K. We performed the measurements of a.c. harmonic resistances using a Keithley 6221 current source and Stanford Research SR830 lockin amplifiers. During the measurements, a constant amplitude sinusoidal current with a frequency of 21 Hz was applied to the devices, and the inphase (0°) first harmonic V_{ω} and outofphase (−90°) secondharmonic V_{2ω} longitudinal voltage signals were measured simultaneously by two lockin amplifiers.
Theoretical modeling
The starting point for our calculations of the carrier density (Fig. 2f) and nonlinear current (Fig. 4a–f) is the electronic band structure of bulk WTe_{2} obtained within DFT. From the abinitio results, we obtain an accurate Wannier tightbinding Hamiltonian representation that describes the electronic structure of this system, and which is subsequently used in our transportrelated calculations. We used the DFTderived tightbinding that has been previously employed to predict Fermi arcs in this material in ref. ^{22}. The observable transport properties are calculated from Boltzmann response theory using the tightbinding Hamiltonian (Supplementary Note 1). In order to isolate analytically the Fermi surface convexity as the underlying cause of the signchange in the nonlinear MR, we also considered the effective k · p Hamiltonian dictated by symmetry for this system, as described in Supplementary Notes 4 and 5. The results shown in Fig. 4g–i have been obtained with this approximated Hamiltonian; all other calculations are based on the full DFTderived tightbinding Hamiltonian.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
The work was partially supported by the National Research Foundation (NRF), Prime Minister’s Office, Singapore, under its Competitive Research Program (CRP award no. NRFCRP12201301). Numerical calculations were performed at the HPC facilities of the NUS Center for Advanced 2D Materials.
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P.H., S.S. and H.Y. designed the experimental study. S.S. and P.H. fabricated devices, P.H. performed the transport measurements and analyzed the data. K.C., J.Y.W, Q.W. and G.E. helped in sample characterization. C.H.H., V.M.P. and H.L. designed the theoretical methodology with C.H.H performing all the numerical calculations. All authors discussed the results. P.H., C.H.H., V.M.P. and H.Y. wrote the manuscript. H.Y. supervised the project.
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He, P., Hsu, CH., Shi, S. et al. Nonlinear magnetotransport shaped by Fermi surface topology and convexity. Nat Commun 10, 1290 (2019). https://doi.org/10.1038/s41467019092088
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DOI: https://doi.org/10.1038/s41467019092088
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