Bulk Fermi surface of the type-II Weyl semimetal candidate NbIrTe$_{4}$

TL;DR

This study investigates the Fermi surface of NbIrTe4, a candidate Weyl type-II semimetal, using quantum oscillation measurements and compares the results with theoretical calculations, revealing topological features and complex magnetotransport behavior.

Contribution

The paper provides experimental evidence supporting the existence of Weyl type-II points in NbIrTe4 through quantum oscillation data aligned with DFT calculations, and analyzes its unconventional magnetoresistance.

Findings

Fermi surface extremal areas match DFT predictions

NbIrTe4 exhibits large, non-saturating magnetoresistivity up to 35 T

Magnetoresistance shows four-fold angular dependence related to Fermi surface topology

Abstract

Recently, a new group of layered transition-metal tetra-chalcogenides were proposed, via first principles calculations, to correspond to a new family of Weyl type-II semimetals with promising topological properties in the bulk as well as in the monolayer limit. In this article, we present measurements of the Shubnikov-de Haas (SdH) and de Haas-van Alphen effects under high magnetic fields for the type-II Weyl semimetallic candidate NbIrTe$_{4}$. We find that the angular dependence of the observed Fermi surface extremal cross-sectional areas agree well with our DFT calculations supporting the existence of Weyl type-II points in this material. Although we observe a large and non-saturating magnetoresistivity in NbIrTe$_{4}$ under fields all the way up to 35 T, Hall-effect measurements indicate that NbIrTe$_{4}$ is not a compensated semimetal. The transverse magnetoresistivity displays a four-fold angular dependence akin to the so-called butterfly magnetoresistivity observed in nodal line semimetals. However, we conclude that its field and this unconventional angular-dependence are governed by the topography of the Fermi-surface and the resulting anisotropy in effective masses and in carrier mobilities.

Tables

Table 1. Table 1: Effective masses of NbIrTe 4 for selected SdH and dHvA frequencies. The pockets are labeled as α 𝛼 \alpha , β 𝛽 \beta , γ 𝛾 \gamma and δ 𝛿 \delta following Figs. ( 3 ) and ( 6 ). The angle θ 𝜃 \theta represents the orientation of the magnetic field where θ = 0 = ∘ 270 ≡ ∘ μ 0 H ∥ a \theta=0{}^{\circ}=270{}^{\circ}\equiv\mu_{0}H\parallel a , 90 ≡ ∘ μ 0 H ∥ b 90{}^{\circ}\equiv\mu_{0}H\parallel b and 180 ≡ ∘ μ 0 H ∥ c 180{}^{\circ}\equiv\mu_{0}H\parallel c , where m ∗ superscript 𝑚 m^{*} is the cyclotron effective mass in units of the free electron mass m 0 subscript 𝑚 0 m_{0} , and F 𝐹 F the SdH/dHvA frequency.

pocket	$\theta ({}_{}{}^{\circ })$	$m^{\ast }\left(m_0\right)$	$F\left(\mathrm{kT}\right)$
$\alpha$	135	0.76	0.178
	180 ($c$)	0.47	0.148
$\beta$	180 ($c$)	0.62	0.289
	225	1.0	0.341
$\gamma$	45	0.46	0.372
	90 ($b$)	1.7	0.299
	125	0.55	0.332
$\delta$	45	2.0	0.729
	90 ($b$)	1.36	0.496

Equations

$${\rho }_{xx}$$

$${\rho }_{xy}$$

Full text

I Introduction

Recently, Weyl fermions have emerged as a heavily studied subject combining key concepts from high energy and condensed matter physics reviews . In a Weyl semimetal, Weyl fermions emerge around the touching points between linearly dispersing valence and conduction bands. Type-I Weyl points correspond to the “conventional” Weyl Fermions of quantum field theory. Weyl semimetals require broken inversion or time reversal symmetry for the Weyl nodes of opposite chirality to separate in $k$-space or to prevent their pairwise annihilation. Insofar, several candidates for type-I Weyl semimetals have been found, most notably the compounds belonging to the TaAs family huang_weyl_2015 ; xu_weyl_2015 ; lv_experimental_2015 ; weng_weyl_2015 . In these compounds angle resolved photoemission spectroscopy (ARPES) and magneto transport experiments were able to reveal a few characteristic signatures of Weyl semimetals like topological Fermi arcs on their surface, the emergence of a controversial negative longitudinal magnetoresistance (NLMR) when magnetic and electric fields are aligned huang_observation_2015 , and the observation of the so-called Weyl-orbits moll . exploring the Fermi arcs.

Weyl type-II fermions are predicted to emerge at the boundary between hole- and electron-pockets resulting from strongly tilted Weyl cones that break Lorentz invariance within the crystal and, therefore, have no equivalents in high energy physics soluyanov_type-ii_2015 . Candidates for Weyl type-II semimetals include non-centrosymmetric materials whose inversion symmetry is broken like the layered orthorhombic transition-metal dichalcogenides MoTe2 and WTe2 sun_prediction_2015 ; wang_mote_2016 , as well as MoP2 and WP2 autes_robust_2016 which do not crystallize in a layered structure. However, type-II weyl points have been predicted to emerge in the ternary tellurides TaIrTe4 and NbIrTe4 koepernik_tairte_2016 ; li_ternary_2017 . In fact, the entire family of the $MM^{\prime}$Te4, where $M$ = Ta or Nb and $M^{\prime}$ = Ir or Rh, is candidate for bulk Weyl type-II semimetallic states and is also predicted to display a topological quantum spin hall insulating phase in the monolayer limit liu_van_2017 . TaIrTe4 was the first representative within this family of materials to be claimed to exhibit a Weyl semimetallic state koepernik_tairte_2016 . It was shown that by including spin-orbit coupling (SOC), TaIrTe4 hosts a minimum of four Weyl points within the first Brillouin zone. Experimental results from quantum oscillations (khim_magnetotransport_2016, ) and ARPES (belopolski_signatures_2017, ; haubold_experimental_2017, ) measurements where able to identify the Weyl points by comparing the measured electronic structure with the DFT calculations. Interestingly TaIrTe4 displays a complex electronic structure that can be tuned by external factors like strain which modifies the number of Weyl points or the location of topological nodal lines zhou_coexistence_2018 .

In this article, we investigate the topography of the Fermi surface of the Weyl type-II semimetallic candidate NbIrTe4 via measurements of the de Haas-van Alphen (dHvA) and the Shubnikov-de Haas (SdH) effects to compare with band structure calculations. Similar to $T_{d}$-MoTe2 and WTe2, NbIrTe4 is a non-centrosymmetric layered compound belonging to the orthorhombic space group Pmn21 mar_metal-metal_1992 , as shown in Fig. 1 (a). Based on DFT calulations, eight Weyl points emerge within the first Brillouin zone in the absence of SOC. After including SOC a total of 16 Weyl points emerge between the topmost valence and the lowest conduction band, which makes the electronic structure of NbIrTe4 more complex than that of its sister compound TaIrTe4. These nodes are located within $142\,\mathrm{meV}$ of the Fermi energy $E_{\mathrm{F}}$ with 8 nodes located in the $k_{z}=0$ plane and other 8 in $k_{z}=\pm 0.2$ planes li_ternary_2017 .

II Methods

Single crystals of NbIrTe4 were grown via a Te flux method. Stoichiometric amounts of elementary Nb ($99.999\%$ Alfa Aesar) and Ir ($99.99\%$ Sigma Aldrich) with excess Te were heated in a sealed quartz ampule up to $1000^{\circ}\mathrm{C}$ and then slowly cooled to $700^{\circ}\mathrm{C}$. After removing the excess Te via centrifugation of the ampoules at $700^{\circ}\mathrm{C}$, we obtained shiny metallic crystals with dimensions up to $5\times 1\times 0.1$ mm3. The inset in Fig. 1(b) shows an image of a typical bar-shaped NbIrTe4 single-crystal. The crystallographic $a$-axis is in general aligned along the longest crystal dimension and the $c$-axis along the shortest. The composition and phase purity of our samples was confirmed by energy-dispersive X-ray spectroscopy (EDS) and X-ray diffraction. Initial resistivity and Hall-effect measurements on NbIrTe4 were performed in a ${}^{4}\mathrm{He}$-cryostat equipped with a $9\,\mathrm{T}$ superconducting magnet (Quantum Design PPMS). Experiments under high magnetic fields up to $35\,\mathrm{T}$ where performed in resistive Bitter magnets at the National High Magnetic Field Laboratory (NHMFL) in Tallahassee using 3He cryostats. AC resistivity measurements were performed on single-crystals using the standard 4-wire technique. Additionally, a capacitive cantilever beam technique was used for magnetic torque measurements.

In order to obtain the electronic band structure of NbIrTe4 and the geometry of its Fermi surface we performed DFT calculations including spin-orbit coupling using the Wien2k package blaha_full-potential_1990 . The Perdew-Burke-Ernzerhof (PBE) exchange correlation functional perdew_generalized_1996 was used in combination with a dense $k$-mesh of $22\times 8\times 8$ $k$-points and a cutoff $RK_{\mathrm{max}}$ of 7.5. The structural parameters were taken from Ref. mar_metal-metal_1992 . The angular dependence of the SdH and dHvA frequencies, which are associated to the extremal cross-sectional areas of the Fermi surface through the Onsager relation, the effective cyclotron masses and the charge carrier densities were calculated using the SKEAF code rourke_numerical_2012 . For the visualization of the crystal structure and of Fermi surface we used the XCrysden kokalj_xcrysdennew_1999 package.

III Results

The layered structure of NbIrTe4 is displayed in Fig. 1(a) and it corresponds to a variant of the WTe2 one as discussed in Ref. mar_metal-metal_1992 . Nb and Ir atoms form zigzag chains along the $a$-direction. When compared to WTe2 the alternation of Nb and Ir results in a doubling of the unit cell along the $b$-axis. Measurements of the resistivity $\rho$ in the absence of an external magnetic field reveal metallic behavior with $\rho(T)$ saturating at low temperatures around a residual resistivity $\rho_{0}\simeq 16$ $\mu\Omega$cm (see, Fig. 1(b)). The residual resistivity ratio (RRR) which is defined here as $\mathrm{RRR}=\rho(300{\text{ K}})/\rho_{0}$ reaches a value of 17 for this particular sample. However, we consistently found RRR values ranging from 15 to 40 across several sample batches. Although these RRR values are not particularly high, when compared to other layered transition metal chalcogenides like MoTe2, the resulting residual resistivites are relatively low with the presence of quantum oscillations confirming that these crystals are of relative high quality. In this manuscript, we include data from 4 NbIrTe4 single-crystals used for resistivity and torque measurements at high fields.

The angular dependence of the transverse magnetoresistivity $\rho(\theta)$ of NbIrTe4 for fields rotating within the $ac$-plane, and for $I\parallel b$-axis, is shown in Fig. 1(d). $\rho(\theta)$ is two-fold symmetric with the additional structure emerging at higher fields resulting from the SdH-effect. Its minimum is observed for fields parallel to the $a$-axis, implying increased inter-layer scattering and a smaller carrier mobility and larger effective masses for fields along this orientation.

For fields rotating in the $bc$-plane, as in shown in Fig. 1(c), $\rho(\theta)$ displays a significantly smaller anisotropy, when compared to fields rotating in the $ac$-plane, reaching its maximum for $\theta\approx 45^{\circ}$ and its minimum for $H\parallel b$. This anisotropy results in a four-fold symmetric “butterfly” shaped angular dependence that is only present under magnetic fields exceeding $\mu_{0}H=10\,T$. Under smaller fields the butterfly disappears and $\rho(\theta)$ becomes maximal for $H\parallel c$. This butterfly shaped magnetoresistance was also observed in the high-$T_{\mathrm{c}}$ superconductors raffy , in magnetic thin films, as well as in the Dirac nodal line semimetal ZrSiS ali_butterfly_2016 . In the case of ZrSiS this behavior was ascribed to a topological phase-transition as a function of field orientation that is inherent to the nodal Dirac line ali_butterfly_2016 although the SOC should gap the nodal lines that are located in close proximity to its Fermi level given that they are associated to symmorphism.

Similarly to TaIrTe4 khim_magnetotransport_2016 , the negative longitudinal magnetoresistance (NLMR) observed in the Weyl type-I monopnictide semimetals huang_observation_2015 ; hu__2016 ; arnold_negative_2016 , which was ascribed to the axial anomaly between Weyl points, is absent in NbIrTe4 for fields and currents along the $a$-axis. For Weyl type-II semimetals the positive longitudinal magneto-conductivity observed in Weyl type-I was originally predicted to depend on the orientation of the external magnetic field relative to wave-vector connecting the Weyl nodes soluyanov_type-ii_2015 ; udagawa . Although, more recently it was claimed to be orientation independent pallab . In NbIrTe4, as well as in TaIrTe4, perhaps the Weyl nodes are located too far away from the Fermi level to lead to charge carriers having a well-defined chirality or perhaps that their electrical transport properties are dominated by the topologically trivial bands.

Figure 2 displays the magneto-resistivity $\rho(\mu_{0}H)$ as a function of the angle $\theta$ for fields rotating within the $ac$ and the $bc$ planes. For $\mu_{0}H\parallel c$-axis SdH oscillations, superimposed onto the magnetoresistive background, become observable when the field exceeds $\mu_{0}H\sim 8$ T. But their amplitude weakens as the field is oriented towards the $a$-direction, limiting the observation of quantum oscillations to $\theta\lesssim 70^{\circ}$ with respect to the $c$-axis. For rotations within the $bc$-plane the SdH oscillations are visible over the entire angular range. Remarkably, over a decade in field the magnetoresistivity can be fit to a single power law where $\rho(\mu_{0}H)\propto H^{\alpha}$ yielding $\alpha=1.42$. Figure S1 in the Supplementary Information file supplemental displays the magnetoresistivity in a log-log scale indicating quite clearly the existence of a single, anomalous power law over an extended range of magnetic fields with this exponent being nearly orientation independent. Notice that this value is quite close to the value $\alpha\simeq 1.5$ extracted for TaIrTe4 khim_magnetotransport_2016 . The first step to address this behavior would be to develop a model within Boltzmann transport theory combining both closed and open orbits, given the geometry of the Fermi surface derived from the calculations which, as shown below, is confirmed by our experiments, including anisotropic effective masses. As discussed by Ref. mr_fermi , such approach is capable of describing non-saturating magnetoresistivity displaying an unconventional power dependence on field in addition to replicating its anisotropy. If conventional transport theory was unable to capture this behavior in a scenario that includes the lack of carrier compensation, one could conjecture that it might bear relation to the existence unconventional quasiparticles.

In order to reveal the topography of the Fermi surface of NbIrTe4, we performed magneto-resistivity measurements as a function of the angle $\theta$ to extract the angular dependence of the SdH frequencies. The results are summarized in Fig. 3, see also Fig. S3 in SI supplemental . To obtain the oscillatory signal, we fit the magnetoresistive background to a polynomial and subsequently subtract it from the experimental curve. The fast Fourier transform of the oscillatory component superimposed onto the resistivity and torque data can be found in the SI material (see, Fig. S3). According to the DFT calculations the Fermi surface of NbIrTe4 consists of two pairs of spin-orbit split electron (labeled as $\gamma$ and $\delta$) and hole-pockets ($\alpha$ and $\beta$) located near the center of the Brillouin Zone (see, Fig. 3(c)). Three of the four pockets ($\alpha$, $\beta$, and $\delta$) are strongly corrugated cylinders aligned along the $k_{z}$ and the $k_{y}$-directions, respectively. The $\gamma$ sheet forms an anisotropic “kidney” shaped pocket. As shown in Fig. 6, for magnetic fields oriented along the $c$-axis we can identify three distinct frequencies $F_{1}$, $F_{2}$ and $F_{3}$. Based on their angular dependence $F_{1}$ and $F_{2}$ can be assigned to the electron pockets and $F_{3}$ to the kidney shaped hole-pocket. Although the lower calculated frequencies associated to the hole pockets are not clearly visible in the experimental data, we can achieve a quite acceptable agreement between calculated and experimental frequencies by lowering the Fermi level by $-17\,\mathrm{meV}$. Only the size of the $\gamma$-pocket is overestimated in the calculations by approximately $30\%$. From the evolution of the Fermi surface with respect to the position of the Fermi energy it is evident that a shift of the $\gamma$ pocket does not affect the Weyl points in NbIrTe4 since they appear at touching points between the $\delta$ and the $\beta$ pockets. Since the Weyl type-II points result from band crossings not associated with the band yielding the $\gamma$ sheet, one can safely state that these nodes are not affected by the accurate position of this band relative to the Fermi level, see band depicted by green line in Fig. S4 within the SI file supplemental .

Further justification for lowering the Fermi level comes from Hall-effect measurements that were performed on a mechanically exfoliated sample with a thickness of $20\,\mathrm{\mu m}$ and lateral dimensions of 1-$2\,\mathrm{mm}$. We extract the charge carrier densities and mobilities of NbIrTe4 from Hall-effect measurements collected between 5 and $100\,\mathrm{K}$ under magnetic fields up to 9 T ($\mu_{0}H\parallel c$-axis) by simultaneously fitting the longitudinal magneto-resistivity $\rho_{xx}$ and the Hall resistivity $\rho_{xy}$ to the two-band model:

[TABLE]

where $n_{\mathrm{e}}$, $n_{\mathrm{h}}$ are the electron and hole carrier densities and $\mu_{\mathrm{e}}$, $\mu_{\mathrm{h}}$ are the electron and hole carrier mobilities, respectively. In this field interval one obtains a reasonable agreement between the experimental data and the fittings, see Fig. 4(a, b). The mobilities increase as the temperature is lowered to $\mu_{\mathrm{e}}\approx 0.12\times 10^{4}\,\mathrm{cm^{2}/Vs}$ and $\mu_{\mathrm{h}}\approx 0.1\times 10^{4}\,\mathrm{cm^{2}/Vs}$ at $T=5\,\mathrm{K}$, while the electron and the hole densities vary little within this temperature interval. From the low temperature moblities we can estimate the classical transport lifetime $\tau_{\mathrm{D}}$ given by the Drude model, $\tau_{\mathrm{D}}=\mu_{\mathrm{e/h}}m^{*}/e=3.5\times 10^{-13}\,\mathrm{s}$ with $\mu_{\mathrm{e/h}}\approx 0.1\,\mathrm{m^{2}/Vs}$ and an effective mass of $m^{*}=0.5m_{\mathrm{e}}$. This result is comparable with the quantum lifetime obtained from SdH oscillations that is related to the Dingle temperature $T_{\mathrm{D}}$: $\tau_{Q}=\hbar/2\pi k_{\mathrm{B}}T_{\mathrm{D}}=2.3\times 10^{-13}\,\mathrm{s}$. As shown in Fig. 4(d), $\mu_{\mathrm{h}}$ is smaller than $\mu_{e}$ for fields parallel to $c$ which can be attributed to open Fermi surface pockets that result in larger effective masses for the hole orbits when compared to the closed electron orbits.

At a temperature of $5\,\mathrm{K}$ we obtain $n_{\mathrm{e}}\approx 0.22\times 10^{20}\,\mathrm{cm^{-3}}$ and $n_{\mathrm{h}}\approx 0.48\times 10^{20}\,\mathrm{cm^{-3}}$. Therefore, the density of holes exceeds the density of electrons by a factor greater than 2. In contrast, the volumes of the individual Fermi surface sheets for the unshifted Fermi level (Fig. 5), would yield nearly equal electron and hole densities, making NbIrTe4 a compensated semimetal. As depicted in Fig. 5 lowering the Fermi level leads to an expansion of the volume of the hole pockets while shrinking the electron pockets without fundamentally changing their shape. Thus lowering the Fermi level leads to a reduction in $n_{\mathrm{e}}$ and to an increase in $n_{\mathrm{h}}$ resulting in a better agreement with the Hall-effect data. Hence, NbIrTe4 is remarkable for not being carrier compensated, and this is consistent with its modest magnetoresistivity, i.e. $\Delta\rho\sim 400$% observed at low $T$s under fields up to $\mu_{0}H=35$ T, while displaying non-saturating magnetoresistivity.

The cyclotron effective mass of a given electronic orbit can be extracted from the temperature damping factor $R_{T}$ in the Lifshitz-Kosevich formula and is given by: $R_{T}=\lambda T/\sinh(\lambda T)$ where $\lambda=2\pi^{2}k_{\mathrm{B}}m^{*}/\hbar eB$, with $m^{*}$ being the cyclotron effective mass. The fast Fourier transform (FFT) of the oscillatory component superimposed onto the torque data for $\mu_{0}H$ nearly parallel to the $c$-axis is shown in the Fig. 6(a). There are at least three distinct frequencies, which we label as $F_{1}$, $F_{2}$ and $F_{3}$. Each correspond to a distinct extremal cross-sectional area of the Fermi surface and belong to a different Fermi surface pocket ($\alpha$, $\beta$, $\gamma$). To extract $m^{*}$ for each individual orbit, we fit $R_{T}$ to the temperature dependence of the amplitude of the peaks observed in the FFT spectra (Fig. 6(b)). As shown in Table 1, $m^{*}$ is anisotropic or depends on sample orientation with respect to $\mu_{0}H$ ranging from 0.46 to twice the free electron mass. These values are comparable to those extracted for TaIrTe4 khim_magnetotransport_2016 .

As seen in Table 1, $m^{*}$ scales with the size of the extremal cross-sectional orbit with the exception of the $\gamma$ pocket, which shows an enhanced effective mass for $H\parallel b$. The presence of open orbits for fields aligned aligned along the crystallographic axes, due to the topography of the Fermi surface is likely to affect the angular dependence of the magnetoresistance and this might explain the butterfly shaped magnetoresistance. As illustrated in Fig. 6(d), the calculated average effective mass is enhanced along high symmetry directions due to the presence of open orbits but depends on the position of the Fermi level. For small shifts ranging from -11 to $-22\,\mathrm{meV}$ the cyclotron effective mass shows a maximum along the $b$ and the $c$ axes but a minimum in between. This would result in a lower carrier mobility $\nu=q\tau/m^{*}$ (where $q$ is the charge of the carrier and $\tau$ the average scattering time) and hence in a smaller magnetoresistivity for fields along the crystallographic $b$- and the $c$-axis, when compared to fields oriented in between both axes which yield closed cyclotron orbits.

IV Summary

In summary, we found that the geometry of the Fermi surface of NbIrTe4 obtained from the DFT calculations and from quantum oscillation measurements are in rather good agreement, if one considers a small shift in the position of the Fermi energy which does not affect the existence of Weyl type-II nodes. Our overall results support the presence of Weyl points in this material although it displays rather conventional transport properties. Furthermore, our results contrast to our previous studies on $T_{d}$-MoTe2 daniel and WP2 Rico , which are also predicted to display a Weyl type-II semimetallic state, but whose experimental Fermi surfaces derived from quantum oscillations can only be captured by DFT after electron and hole bands are independently displaced towards higher and lower energies, respectively. For both compounds this ad-hoc procedure would suppress their Weyl points, in contrast to their robustness in NbIrTe4. Both quantum oscillations and Hall-effect measurements indicate that NbIrTe4 is not a compensated semimetal and that the unconventional four-fold anisotropy of its angular magnetoresistivity is governed by the topography of its Fermi surface and related anisotropy in effective masses. The good agreement between bulk measurements and band structure calculations imply that the electronic properties of NbIrTe4, and hence its topological character, are well-captured by Density Functional Theory calculations. More importantly, since this compound is exfoliable and predicted by DFT to display a quantum-spin-Hall insulator state in the monolayer limit liu_van_2017 , as recently found for WTe2 pablo , our conclusions convey that it would be important to explore edge conduction and the effect of a gate voltage in heterostructures containing monolayers of NbIrTe4.

V Acknowledgments

We thank S. Sur and V. Quito for helpful discussions. This work was supported by DOE-BES through award DE-SC0002613. JYC acknowledge NSF DMR-1700030 for partial support. The NHMFL is supported by NSF through NSF-DMR-1644779 and the State of Florida.