Symmetry-enforced Fermi degeneracy in topological semimetal RhSb$_3$

TL;DR

This study investigates RhSb$_3$, a topological semimetal with symmetry-enforced Fermi degeneracy, revealing unique electronic properties such as linear magnetoresistance, high mobility, and a non-zero Berry's phase, supported by experiments and calculations.

Contribution

It demonstrates RhSb$_3$ as a new class of topological semimetals with symmetry-protected Fermi degeneracy and detailed experimental characterization of its electronic structure.

Findings

Linear magnetoresistance approaches 200 at 60 T

Quantum oscillations indicate high Fermi velocity and mobility

Effective mass as low as 0.018 times electron mass

Abstract

Predictions of a topological electronic structure in the skutterudite TPn$_3$ family (T=transition metal, Pn=pnictogen) are investigated via magnetoresistance, quantum oscillations and angle-resolved photoemission experiments of RhSb$_3$, an unfilled skutterudite semimetal with low carrier density. Electronic band structure calculations and symmetry analysis of RhSb$_3$ indicate this material to be a zero-gap semimetal protected by symmetry with inverted valence/conduction bands that touch at the $\Gamma$ point close to the Fermi level. Transport experiments reveal an unsaturated linear magnetoresistance that approaches a factor of 200 at 60~T magnetic fields, and quantum oscillations observable up to 150~K that are consistent with a large Fermi velocity ($\sim 1.3\times 10^6$ m/s), high carrier mobility ($\sim 14$ m$^2$/Vs), and the existence of a small three dimensional hole pocket. A very small, sample-dependent effective mass falls to values as low as $0.018(2)$ of the bare electron mass and scales with Fermi wavevector. This, together with a non-zero Berry's phase and location of the Fermi level in the linear region of the valence band, suggests RhSb$_3$ as representative of a new class of toplogical semimeals with symmetry-enforced Fermi degeneracy at the high symmetry points.

Equations

$$\Delta {\sigma }_{xx}(T,B)=A(T,B)\cos \{2\pi [(F/B)-1/2+\beta +\delta ]\},$$

Full text

Following the discovery of topological insulators TIReview1 ; TIReview2 , new classes of topological materials such as Dirac semimetals Young_2012 , Weyl semimetals Wan_2011 ; Fang_2012 , nodal line and nodal chain semimetals NodalChain ; NodalLine , have developed considerable interest. In Dirac systems, linearly dispersing valence and conduction bands touch at discrete points of four-fold degeneracy in the Brillouin zone (BZ), giving Dirac nodes protected against gap formation by crystal symmetry. Cd3As2 and Na3Bi were the first theoretically predicted Dirac semimetal candidates A3Bi ; Cd3As2-1 , later confirmed by experiments Borisenko_2014 ; Neupane_2014 ; Liu2014 ; Jeon_2014 . By breaking either inversion or time reversal symmetry, a Dirac semimetal can be tuned to a Weyl state where the nondegenerate linear touchings of the bulk bands come in pairs Wan_2011 ; Fang_2012 , confirmed by recent experiments on materials such as the TaAs family (Type-I) TaAs-ARPES1 ; TaAs-ARPES2 . Recent studies have shown the existence of other exotic toplogical semimetal types, such as type-II Weyl semimetals like WTe2 and MoTe2 WTe2-Weyl ; MoTe2-ARPES , nodal-line/nodal-chain semimetals CaAgAs_2016 ; IrF4_2016 ; ZrSiS_2016 , and triple fermions or beyondTriple-2 ; TopologicalQuantumChemisty ; EightFold ; TMDatabase ; TMDatabasewebsite . The topologically nontrivial band structure in such materials hosts unusual electronic states and exotic physical properties, including the so-called chiral anomaly phenomena and associated negative magnetoresistance (MR), non-local transport and the quantum anomalous Hall effect Liang_2014 ; Narayanan_2015 ; Wang2013a ; Shekhar_2015 ; ZrTe5_Chiral ; TaAs-Jia ; TaAs-Chen ; NbAs-Luo . Interestingly, most Dirac and Weyl semimetals exhibit high mobilities and extremely large values of MR  Liang_2014 ; Shekhar_2015 ; Narayanan_2015 ; TaAs-Jia ; TaAs-Chen ; NbAs-Luo ; ZrSiS-MR : carrier mobilities in Cd3As2, TaAs, and WTe2 approach $100$ m2/Vs, and MR scales of up to $\sim 5000$ are observed in 9 T fields. Extreme MR is believed to arise from the lifting of the high mobility of the Dirac/Weyl node protected by the (crystal or time reversal) symmetry or large fluctuations in the mobility due to the disorder effect of the Dirac quasiparticles Liang_2014 ; Narayanan_2015 . Large MR also observed in some other semimetals such as LaSb and the NbSb2 family LaSb_Tafti ; NbSb2-Wang ; NbAs2_Shen ; TaAs2_Luo related to the topological band properties LaBi_ARPES3 ; LaBi_ARPES2 ; LaBi_ARPES1 ; LaSb_ARPES1 ; NbSb2_Z2 ; NbSb2_HiddenWeyl .

Although extensive studies have been reported, unique topological semimetal families are still rare. Seminal theoretical work by Singh et al. on the TPn3 (T=transition metal, Pn=pnictogen) skutterudites that predated the topological revolution identified the possibility of an unusual quasi-linear band dispersion structure in the cubic unfilled skutterudite CoSb3 CoSb3-PRL . CoSb3 is a well known thermoelectric small gap semiconductor Tang_2015 ; Singh_1994 , that, while topologically trivial, was predicted to be tunable through a topological quantum critical point via displacement of the Sb sublattices. Such a displacement was proposed to decrease gap size toward a critical point where the gap closes and massless (Dirac or Weyl) bands appear CoSb3-PRL ; CoSb3-PRB . With further displacement, spin-orbit coupling would open a gap between inverted bands, realizing a non-trivial topological state. Similar behavior was also proposed in the theoretical compound IrBi3, where band inversion between Ir-$d$ and Bi-$p$ orbtitals was predicted to occur IrBi3 . In this work, we report the first exploration of the intermediate member RhSb3 using several experimental methods to identify a new class of topological semimetals with symmetry-enforced Fermi degeneracy at the high symmetry points, that is, enforced semimetals with Fermi degeneracy (ESFD) TMDatabase ; TMDatabasewebsite , in this skutterudite series. Together with observations of very large MR and high carrier mobility values, quantum oscillations studies of the effective mass and non-trivial Berry’s phase, as well as the angle-resolved photoemission spectroscopy (ARPES) experiments, are consistent with first-principle calculations of the electronic structure indicating a non-trivial topology is present in this family.

The TSb3 system with T=Co, Rh, and Ir, crystallizes in the symmorphic unfilled skutterudite structure (space group Im$\bar{3}$) with cubic Bravais lattice as shown in the inset of Fig. 1(a) and Fig. S1 in Supplementary Materials. Without SOC, for the space group Im$\bar{3}$, the $\Gamma$ point in the first Brillouin zone has a point group symmetry of $T_{h}$. In CoSb3, the highest valence band mainly derived from the $p$-orbitals of Sb is found to form an irreducible representation $\Gamma_{1}^{-}$ of the $T_{h}$ point group, while the three lowest conduction bands dominated by the $d-e_{g}$ orbitals of the (Co) transition metal atoms form an irreducible representation $\Gamma_{4}^{+}$. The CoSb3 band structure is highly symmetric, with a single set of occupied and unoccupied bands touching at the high symmetry $\Gamma$ point. Turning on SOC lifts the degeneracy of these states and induces a gap of $\sim$50-120 meV between the valence band with an irreducible representation $\Gamma_{5}^{-}$ and the conduction bands with irreducible representations $\Gamma_{5}^{+}$ and $\Gamma_{6}^{+}+\Gamma_{7}^{+}$, for the $m3$ double point group  Singh_1994 ; CoSb3-PRL ; Koga_2005 ; wzjcode , as shown in Fig. 1(a) CoSb3-PRL .

In RhSb3, the stronger SOC in Rh atoms pushes the band $\Gamma_{5}^{-}$ up as shown in Fig. 1(b). The energy difference $\eta=E_{\Gamma_{6}^{+}+\Gamma_{7}^{+}}-E_{\Gamma_{5}^{-}}$ indicating the strength of the band inversion, becomes negative. This implies the band inversion happens in RhSb3 at the $\Gamma$ point. The SOC also lifts a degeneracy and opens a $\sim 20$ meV gap between the single band $\Gamma_{5}^{+}$ and the double-degenerated bands $\Gamma_{6}^{+}+\Gamma_{7}^{+}$, in Fig. 1(c). Further, the overlap of the valence band $\Gamma_{5}^{-}$ and the conduction bands $\Gamma_{5}^{+}$ and $\Gamma_{6}^{+}+\Gamma_{7}^{+}$ induces a strong hybridization between the $p$ orbitals of Sb atoms and the $d$ orbitals of Rh atoms which is evidenced by the calculated negative $\xi=E_{\Gamma_{6}^{+}+\Gamma_{7}^{+}}-E_{\Gamma_{5}^{+}}$ in Fig. 1(d). Correspondingly, in the highest valence band, the part far away from the $\Gamma$ point is mainly contributed from the $p$-orbitals of Sb atoms. However, the weight of Rh-$4d$ orbitals (indicated by the radii of circles in the band line of Fig. 2(a,b)) increases when approaching the Fermi level and becomes dominant. This clearly indicates a band inversion between Rh$-d$ and Sb-$p$ orbitals. The highest valence band and the lowest conduction band form the irreducible representations $\Gamma_{6}^{+}+\Gamma_{7}^{+}$, and the band $\Gamma_{5}^{-}$ sits above all these bands. However, there is no influence on the double-degenerated bands $\Gamma_{6}^{+}+\Gamma_{7}^{+}$ at $\Gamma$ because the double degeneracy is enforced by the $m3$ double point group. Therefore the Fermi energy of RhSb3 is pinned to the doublet $\Gamma_{6}^{+}+\Gamma_{7}^{+}$, and is protected by symmorphic crystalline symmetry and time reversal symmetry CoSb3-PRB . As shown in the band structure in Figs. 2(b), a Dirac-like quasi-linear dispersion persists over a wide energy range, with a more parabolic dispersion appearing in a very small energy window of $\sim 20$ meV close to the Fermi level as shown in Fig. 2(b). The resultant calculated Fermi surface is an extremely small isotropic hole pocket centered at the $\Gamma$ point (Fig. 2(c)). Together, the band inversion and degeneracy near the Fermi level point to RhSb3 as a new type of ESFD. The ESFD is easily tuned to a topological insulator by symmetry breaking such as a uniaxial strain in the $z$ direction. In other words, an infinitesimal symmetry breaking perturbation can lead it to a topological insulating phase, which is confirmed by the Fu-Kane $Z_{2}$ invariants ($\nu_{0}=1,\nu_{1}=\nu_{2}=\nu_{3}=0$) in the calculation with uniaxial strain (shown in Fig. S2(b) in Supplementary Materials).

To evaluate and confirm the non-trivial topology of these bands, we performed both ARPES and quantum oscillations experiments. Photon energy-dependent ARPES measurements confirm the presence of a linearly-dispersive three-dimensional hole pocket around $\Gamma$. The sample was cleaved to expose the (001) surface. Assuming a 10 eV inner potential, at 140 eV photon energy the momentum probed along the $c$-axis corresponds to $k_{z}\sim 4.5\times 2\pi/c$ ($\pi/c=\Gamma-H$). As a consequence of the body-centered cubic structure, the $\Gamma$ points from adjacent surface-projected BZs are characterized by a $\pi/c$ offset in $k_{z}$. For this reason, hole-like Fermi surface pockets can be observed in Fig. 3(a) only around the $\Gamma$ points of the second BZ. Furthermore, due to matrix element effects, when using linear horizontal polarization (p), the second BZ hole pockets are detected only along the $k_{x}=0$ Å*-1* line while, with linear vertical polarization (s), those same pockets are observed exclusively along the $k_{y}=0$ Å*-1* line (See Fig. S3 of Supplementary Materials).

The resultant ARPES band mapping along $k_{x}$, $k_{y}$ and $k_{z}$ directions is presented in Fig. 3(b). As a consequence of matrix elements effects and to maximize the photoemission intensity from the hole pocket, these three cuts have been acquired in different regions of momentum space using different photon energies and incident polarizations. The left-hand side and central panels of Fig. 3(b) present the band mapping along the $k_{x}$ and $k_{y}$ directions (yellow and white dashed lines in (a), respectively). By fitting the momentum distribution curves in the left panel, we estimate a Fermi velocity of $(0.6\pm 0.2)\times 10^{6}$ m/s ($4\pm 1.2$ eV Å). In order to verify the three-dimensional nature of the detected pocket, photon-energy scans were performed to probe the $k_{z}$ dependence. As shown in Fig. 3(b) right panel, the photoemission intensity at $(k_{x},k_{y})=(-0.57$ Å${}^{-1},0$ Å${}^{-1})$ (slightly off the $\Gamma$ point) changes as a function of energy, qualitatively defining a nearly isotropic three-dimensional cone-like dispersion for the hole pocket. This is in good agreement with band structure calculations, and for comparison is in contrast to observations of Cd2As3, where the Fermi velocity is markedly anisotropic Liu2014 . Furthermore, the field angular dependence of the frequency of quantum oscillations in RhSb3 crystal (shown in Fig. S4 of Supplementary Materials) exhibits very little variation with field angle orientation, confirming the case for a nearly spherical bulk three-dimensional Fermi pocket.

Magnetotransport and Shubnikov-de Haas (SdH) quantum oscillations experiments were performed on four representative samples. Magnetic fields applied along the principle axis in a transverse configuration significantly enhance the resistivity of RhSb3. The MR ($=(\rho_{xx}(B)-\rho_{xx}(0))/\rho_{xx}(0)$) is linear in the high field region as evidenced by the field derivative of MR at different temperature respectively for Sample D shown in Fig. S8 in the Supplementary Materials, and does not show any sign of saturation up to a 64 T pulsed field (Fig. 4(a)) but exhibits quantum oscillations above 10 T (also observed in Hall resistivity, Fig. 4(b)), and continues to increase above the quantum limit of $\sim 35$ T. The MR ratio is close to the reported value ($10^{3}\sim 10^{6}$) observed in both the Dirac material Cd3As2 and the TaAs family of Weyl materials Liang_2014 ; Shekhar_2015 ; Narayanan_2015 ; TaAs-Jia ; TaAs-Chen ; NbAs-Luo ; ZrSiS-MR As shown in Fig. 4, the MR has a strong temperature dependence, driving a semi-metallic character in zero field toward a semiconductor-like behavior with a saturating resistivity at the lowest temperatures. While the absolute MR amplitude varies among different samples and is dependent on the residual resistivity ratio of the crystals, all crystals show a crossover from semimetallic character to semiconductor-like behavior driven by magnetic field (see Figs. S5 and Fig. S6 in Supplementary Materials for more details).

Fig. 4(b) presents the Hall resistivity $\rho_{xy}$ of RhSb3 as a function of field at different temperatures. In fields below 6 T, $\rho_{xy}$ is linear and nearly temperature-independent, while showing large quantum oscillations at higher fields that degrade with increasing temperature. The carrier density $n$ of sample D is calculated by linear fits of the low-field $\rho_{xy}$, yielding a value $n\sim 1.1\times 10^{18}$ cm*-3* at low temperature and only $\sim 10\%$ variation upon increasing temperature as shown in the inset of Fig. 4(b). This is consistent with the zero-gap band structure depicted in Fig. 2. From the carrier density $n$ and the zero-field resistivity $\rho_{xx}=18~{}\mu\Omega$cm, the classical mobility $\mu_{c}=1/\rho_{xx}ne$ is calculated to approach 14 m2/Vs at 1.8 K, nearly that of graphene, Cd3As2 and TaAs/TaP, which fall in the range 1-100 m2/Vs.

In addition to MR variations, SdH oscillations measurements performed on four different samples indicate a tunability of the oscillation frequency, and hence the carrier density and corresponding chemical potential variation. Quantum oscillations in conductivity are given by

[TABLE]

with the non-oscillating amplitude $A(T,B)=e^{-2\pi^{2}k_{B}T_{D}/\hbar\omega_{c}}\frac{2\pi^{2}k_{B}T/\hbar\omega_{c}}{\sinh(2\pi^{2}k_{B}T/\hbar\omega_{c})}$, Dingle temperature $T_{D}$, cyclotron frequency $\omega_{c}=eB/m^{*}$ with effective mass $m^{*}$, phase factor $-1/2+\beta+\delta$ and SdH frequency $F$ of the oscillation corresponding to the cross-section of Fermi surface defined by the cyclotron orbits OscillationBook . For a trivial parabolic dispersion, $\beta$=0 and therefore the Berry’s phase is zero, while $\beta$=1/2 for a Dirac dispersion giving a Berry’s phase of $\pi$. $\delta$ is a phase shift resulting from the curvature of the Fermi surface in the third direction, taking a value of $\delta=0$ ($\pm\frac{1}{8}$) for a two-dimensional (three dimensional) Fermi surface.phase-shift1 ; phase-shift2 ; phase-shift3 ; phase-shift4 The frequency $F$ is given by the Onsanger relation $F=\frac{\hbar\pi k_{F}^{2}}{2\pi e}$ with $k_{F}$ being the Fermi wave vector in the spherical Fermi surface approximation.

As evident in Fig. 4, the SdH oscillations of sample D have a very small frequency, which is 33 T as extracted from FFT analysis of the background-subtracted resistivity (see SM Fig. S6 for FFT spectra of all samples). This value corresponds to wave vector $k_{F}=0.04(5)~{}\AA^{-1}$ and a very small carrier density $n_{SdH}=0.8(1)\times 10^{18}$ cm*-3*, which is consistent with Hall resistivity results. The other three crystals also exhibit SdH oscillations (see Fig. S7 in Supplementary Materials) with varying amplitudes. FFT analysis yield a relatively large variation in frequencies: 71 T, 60 T and 43 T for samples A, C and E, respectively. Moreover, oscillations observed in Sample D appear to decay very slowly with increasing temperature, still being observable at 150 K. Interestingly, the amplitude of oscillations appears to be suppressed at different temperatures for each sample. While oscillations are still evident at 150 K for sample D (Fig. 4), they disappear near 30 K, 50 K and 100 K for samples A, C and E, respectively. This suggests that samples with smaller frequency (or carrier density) have charge carriers with systematically smaller effective masses. The decaying of the amplitude of SdH oscillation with temperature is described by the Lifshitz-Kosevich (LK) formula $A(T,B)$ given by Eqn.(2). As shown in Fig. 4(d), tracking the FFT amplitudes of all four samples as a function of temperature indeed shows a systematic trend. Fitting to the LK formula gives effective mass ratios $m^{*}/m_{e}$ of 0.102(8), 0.066(7), 0.033(4) and 0.018(2) for samples A, C, E and D, respectively, in descending mass order.

The effective mass in RhSb3 is extremely light, being rivaled only by InSb, which has an effective mass $\sim 0.014m_{e}$ that yields observable oscillations up to 175 K InSb1 ; InSb2 . The effective mass in RhSb3 is also in fact comparable to those observed in typical Dirac ($0.023m_{e}$ in Cd3As2 Liang_2014 ; Narayanan_2015 ) and Weyl ($0.033m_{e}$ in NbAs NbAs-Luo , $0.076m_{e}$ in NbP Shekhar_2015 and $0.15m_{e}$ in TaAs Huang2015 ) semimetals. However, in RhSb3 the systematic variation of $m^{*}$ with $F$ is indicative of a departure from the expectations for a parabolic dispersion. The effective mass, defined as $m^{*}=\hbar k_{F}/v_{F}$ should be constant for a parabolic band with different carrier densities since the Fermi velocity $v_{F}=\frac{1}{\hbar}\frac{\partial\varepsilon}{\partial k}|_{k_{F}}\propto k_{F}$ with $\varepsilon$ being the band energy. However this is not true for Dirac fermions with a linear dispersion, where $v_{F}$ is a constant. For instance, in graphene it is expected that $m^{*}$ is proportional to the square root of carrier density since $k_{F}=(\pi n_{s})^{1/2}$, which was indeed confirmed by several experiments Graphene_RMP ; Graphene_Novoselov_2005 Similar phenomenon has also been shown in Cd3As2 Narayanan_2015 . In RhSb3, the relation between $m^{*}$ and $k_{F}$ follows the equation above very well, as shown for all four samples in Fig. 4(d), except for an intercept which is possibly due to the breakdown of the semiclassical transport theory close to the quantum limit. This giving a constant $v_{F}=1.3\times 10^{6}$ m/s that is close to the ARPES value and also to that of Cd3As2 ($\sim 4\times 10^{6}$ m/s) and NdP ($\sim 4.8\times 10^{5}$ m/s).

The observation of a constant Fermi velocity with varying carrier concentration in RhSb3 directly confirms that the bulk carriers have a linear dispersion very close to the degeneracy point, and the extreme values of $m^{*}$ and $v_{F}$ are considered to be responsible for the high mobility values. The topological nature of this band structure can be further explored by analysis of the phase factor of the quantum oscillations as indicated in Eq. (1). Experimentally, this phase shift in the semi-classical regime can be obtained from an analysis of the relation between the Landau level (LL) index $N$ and energy, widely represented in a so-called fan diagram of integer $N$ plotted as a function of inverse field $1/B$. The slope of such a plot is dependent on the oscillation frequency $F$ and the y-axis intercept yields the Berry’s phase $\beta$, in units of $2\pi$. For a topologically trivial material with parabolic dispersion, an intercept of zero is expected, while that with a chiral Dirac dispersion would possess finite Berry curvature and therefore a Berry’s phase of value $\pi$.

Fig. 5(a) presents the oscillatory component of the conductivity $\Delta\sigma_{xx}$ of samples C, D and E as a function of $1/B$. Assigning the minima and maxima of $\Delta\sigma_{xx}$ to integer ($N$) and half-integer ($N+1/2$) indices (for the oscillation in high field part, the position of minima is defined as an average of two sub-minima due to the spin splitting), respectively, yields the LL fan diagram as shown in Fig. 5(b), demonstrating the variation in slopes that correspond to the different carrier densities of each sample. A linear extrapolation of $N$ versus $1/B$ to the infinite field limit for each sample yields finite intercepts of $0.352(4),0.381(2)$ and $0.373(6)$ for samples C, D and E, respectively. To properly extract the Berry’s phase values, one must consider the phase shift due to geometry. Taking the $\delta=-\frac{1}{8}$ phase shift due to the spherical geometry of the Fermi surface (the hole pocket) of RhSb3 into account, the measured intercept values are very close to the adjusted value $3/8$ corresponding to the geometry-adjusted phase shift of $1/2+\delta$, yielding a phase factor $\beta=\gamma+\frac{1}{8}$ for a Berry’s phase of $0.92(1)\pi$, $1.02(1)\pi$ and $0.98(1)\pi$ for samples C, D and E, respectively. Because the linear extrapolation of the LL indices is a simplistic fit of the LL spectrum, we have implemented a global fitting routine to further confirm the nontrivial Berry’s phase in this study. Using Eq. 1, we fit $\Delta\sigma_{xx}$ of samples C, D and E using the ”Bumps” code (see Methods section) Bumps , incorporating a Zeeman splitting term $F\pm\delta_{F}$ to capture the observable splitting to yield well-converging fits (shown in SI materials Figs. S9 and S10) to the experimental data as shown in Fig. 5(a). The Berry’s phases extracted from this global fitting procedure are $0.87(1)\pi$, $0.96(1)\pi$ and $0.81(1)\pi$ for samples C, D and E, respectively, consistent with the LL index extrapolations and therefore unequivocally implying the presence of a non-trivial topology and the existence of Dirac quasiparticles in the bulk of RhSb3. While the energy resolution is not enough in both experiment (ARPES) and theoretical calculation to resolve the exact nature of the dispersion at the degeneracy point (i.e. linear or quadratic), the lack of any observable curvature in the fan diagram suggests that the dispersion is very linear and Dirac-like ando , prompting further theoretical consideration of the exact nature of topology in this new class of zero-gap topological semimetals.

I Methods

Single crystals of RhSb3 were synthesized by a high-temperature self-flux method. X-ray diffraction data were taken at room temperature with Cu Kα ($\lambda=0.15418$ nm) radiation in a powder diffractometer. Electrical transport measurements up to 14 T were conducted on polished samples in a commercial cryostat applying the conventional four-wire contact method. High field MR and Shubnikov-de Haas (SdH) oscillation were measured at the National High Magnetic Field Laboratory (NHMFL) Tallahassee DC Field Facility up to 35 T, using a one-axis rotator in a He-3 cryostat with temperature range 0.35 K $\sim$ 70 K, and also at the NHMFL Los Alamos pulsed field facility up to 65 T. All measurements were performed with electrical current flowing along the [100] direction. Magnetic field direction was always kept perpendicular to current, but rotated from perpendicular to (100) face ($\theta=0^{o}$) to direction along (100) face ($\theta=90^{o}$). The longitudinal conductivity $\sigma_{xx}$ is derived through tensor conversion from the longitudinal resistivity $\rho_{xx}$ and the Hall resistivity $\rho_{xy}$ by $\sigma_{xx}=\frac{\rho_{xx}}{\rho_{xx}^{2}+\rho_{xy}^{2}}$.

The global fitting and uncertainty analysis for the quantum oscillation in the longitudinal conductivity $\sigma_{xx}$ for BaAl4 single crystals was performed using a Broyden-Fletcher$-$Goldfarb-Shanno (BFGS) optimizer and a Markov chain Monte Carlo (MCMC) method to compute the joint distribution of parameter probabilities and to find the global minimum, implemented in the “Bumps” code.Bumps Bumps is a set of free and public routines for complicated curve fitting, uncertainty analysis and correlation analysis from a Bayesian perspective. Bumps provides uncertainty analysis which explores all viable minima using the MCMC method and finds confidence intervals on the parameters based on uncertainty from experimental errors. For full uncertainty analysis, bumps further uses a random walk to explore the viable parameter space near the minimum, showing pair-wise correlations between the different parameter values. The 2D correlation plots indicate the correlation relationship between multiple parameters in the fitting function.

ARPES measurements were performed at beamline 7.0.2.1 of the Advanced Light Source (ALS) at Lawrence Berkeley National Laboratory. The sample was cleaved and kept at 80 K at a base pressure of $5\times 10^{-11}$ Torr. Photoemitted electrons were detected with a Scienta R4000 analyser, for photon energy ranging from 130 to 170 eV. The energy and the angular resolution were better than 50 meV and 0.2o, respectively.

First principles electronic structure calculations were performed within the full-potential linearized augmented plane wave (LAPW) method wien2k1 implemented in WIEN2k package, without or with spin-orbit coupling.wien2k2 The general gradient approximation (GGA) of Perdew et al.,gga was used for exchange-correlation potential. The LAPW sphere radius were set to 2.5 Bohr for all atoms. The converged basis corresponding to $R_{min}k_{max}=7$ with additional local orbital were used, where $R_{min}$ is the minimum LAPW sphere radius and $k_{max}$ is the plane wave cutoff. Lattice parameters were obtained from refinement of the x-ray diffraction pattern obtained from powdered single crystals. In the calculation with spin-orbit coupling, the modified Becke-Johnson (mBJ) exchange potential was combined with the GGA correlation.mBJ It can effectively mimic the behavior of orbital dependent potential around the band gap, so is expected to obtain the accurate position of states near the band edge and the band order, which are the keys to determine the band inversion and the band topology. For high-symmetry points, the eigenvalues (irreducible represenations) of all symmetry operators have been computed from the wave functions for each band using the open-source code vasp2trace wzjcode , referring to the character tables in the Bilbao Crystallographic Server (BCS) server ; crystallographicserver .

II acknowledgments

The authors acknowledge useful discussions with A. Bernevig, Z.J. Wang, and J. Sau. This work was supported by the Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant GBMF9071. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR-1157490 and the State of Florida. This research was undertaken thanks in part to funding from the Max Planck–UBC Centre for Quantum Materials and the Canada First Research Excellence Fund, Quantum Materials and Future Technologies Program. The work at UBC was supported by the Killam, Alfred P. Sloan, and Natural Sciences and Engineering Research Council of Canada’s (NSERC’s) Steacie Memorial Fellowships (A.D.); the Alexander von Humboldt Fellowship (A.D.); the Canada Research Chairs Program (A.D.); and the NSERC, Canada Foundation for Innovation (CFI), and CIFAR Quantum Materials.