Experimental nuclear quadrupole resonance and computational study of the structurally refined topological semimetal TaSb$_2$

The local electric field gradients and magnetic dynamics of TaSb$_2$ have been studied using $^{121}$Sb, $^{123}$Sb, and $^{181}$Ta nuclear quadrupole resonance (NQR) with density functional theory (DFT) calculations using XRD-determined crystal structures. By measuring all structurally expected thirteen NQR lines, the nuclear quadrupole coupling constant ($\nu_Q$) and asymmetric parameter ($\eta$) for Ta, Sb(1), and Sb(2) sites were obtained. These values are all in good agreement with the presented DFT calculations. Principal axes of the electric field gradients was determined for a single-crystal sample by measuring the angular dependencies of NMR frequency under a weak magnetic field. The unusual temperature dependence of $\eta$(T) of Sb(2) hints at the suppressed thermal expansion along the $a$-axis. Spin lattice relaxation rate ($1/T_1T$) measurements reveal an activated-type behavior and an upturn below 30 K. Neither the low temperature upturn nor the high temperature activation type behaviors are reproduced by the calculated $1/T_1T$ based on the calculated density of states (DOS). On the other hand, the agreement between the calculated DOS and specific heat measurements indicates that the band renormalization is small. This fact indicates that TaSb$_2$ deviates from the simple semimetal scenario, and magnetic excitations are not captured by Fermi liquid theory.


I. INTRODUCTION
Topological insulators and semimetals are receiving tremendous attention in the solid state research community [1][2][3][4][5][6][7].A spontaneous symmetry breaking via spin-orbit interactions generates a new unconventional phenomenon beyond the Fermi Landau theory.This required a completely new concept of theoretical description and the introduction of new fermions, the so-called Dirac or Weyl fermions.Dirac and Weyl semimetals have unique electronic structures with linear dispersion in momentum space.The direct experimental proof of this fascinating new physics was achieved by angle resolved photoemission spectroscopy [8,9].Indirectly one sees signatures of these topological states formed out of Weyl or Dirac fermions in measurement methods such as resistance (magnetotransport, Shubnikov-de Haas oscillations [10][11][12][13][14]), magnetization (de Haas-van Alphen oscillations [15][16][17]) or also in the optical conductivity [18][19][20].The first Weyl semimetals studied in detail are found among the monopnictides TMPn(TM=Ta,Nb,Pn=P,As) [21].These systems clearly show the presence of the Weyl nodes near the Fermi level [22].After the discovery of superconductivity in Dirac/Weyl monopnictides (TlSb, NbP) [23,24], Weyl dichalcogenides (MoTe 2 ) [25] and Dirac dipnictides (CaSb 2 ) [26,27], we extend our studies towards the dipnictides TMPn 2 (TM=Nb, Ta, Pn= P, As, Sb).In contrast to the monopnictides, TMPn 2 exhibits a centrosymmetric crystal structure.NbAs 2 and TaSb 2 show a negative magnetoresistance [28,29], similar to that observed in TaAs and NbP [13,30].For the latter compound, this has been claimed to originate from the chiral anomalies in the Weyl fermion.Theoretically, the band structure calculation shows no Weyl fermions in TaSb 2 and there are electron and hole bands which are not crossing each other [31,32].Gresch et al., theoretically proposed that the effect of the magnetic field induces the Weyl fermions in TMPn 2 [32], that can explain the appearance of negative magnetoresistance.However, the question arises what characteristics of the band structure are formed under a zero magnetic field.A key to understand the physics of these compounds is to examine whether unique topological characters are present under a zero magnetic field.
Using 181 Ta nuclear quadrupole resonance (NQR) on TaP, we have succeeded for the first time in detecting the excitations of the Weyl fermions near the Fermi level [33].In general, the nuclear magnetic resonance (NMR) method has been successfully applied to Dirac/Weyl fermion material under a magnetic fields [34][35][36][37][38], and these experimental results are consistent with theory [39,40].Ta based topological semimetals are particularly suitable as Ta has one of the largest quadrupole moments which facilitates the NQR measurements.Moreover, the NQR method allows us to explore electronic states under a zero magnetic field.In particular, the spin-lattice relaxation rate (1/T 1 ), reflecting the density of states (DOS) in non-magnetic metals and semimetals, has been successfully applied to several Weyl semimetals [33,38].
The aim of this paper is to explore the local electric field gradient and the magnetic excitations in TaSb 2 using mainly the NQR method.Special attention was given to the quality of both the polycrystalline and the single crystal samples used.Especially in semimetals, it has been known that stoichiometry plays a crucial role.For this reason, detailed investigations by energy dispersive x-ray spectroscopy (EDXS) and structural investigation by x-ray scattering were carried out.We conducted a comprehensive NQR study on the two Sb sites and the Ta site.The temperature dependence of the electric field gradient (EFG) and 1/T 1 at the two Sb sites and the Ta site have been studied in great detail.Here, we have also conducted single crystal NMR measurements under a weak field (WFNMR) to check the site assignment of the observed NQR lines and to obtain the EFG tensor experimentally as well.The experimental NQR parameters thus obtained in our detailed study are in rather good agreement with the results of the theoretical calculation based on density functional theory (DFT).

II. CRYSTAL STRUCTURE AND SAMPLE PREPARATION
TaSb 2 crystallizes in the centrosymmetric space group C2/m (NbSb 2 -type) and contains two inequivalent Sb(1) and Sb(2) sites, and one Ta site.As shown in Fig. 1, the skeleton of the structure can be viewed as separate units composed by two Ta and two Sb(1) atoms linked by P σ bonded Sb(2) and Sb(2) dumbbells along the a-axis (see Fig. 1).These dumbbells represent a covalent bond between two Sb(2) site.The dumbbell can accommodate four electrons, resulting in an average valence of -2 for the Sb(2) site, while the Sb(1) site has a valence of -3.Thus the Zintl rule is satisfied.Therefore, it is easily inferred that the electronic and photonic dispersions are highly anisotropic with the a-axis being a special axis.
Single crystals were grown by a chemical vapor transport reaction using iodine as a transport agent.TaSb 2 has first been synthesized by a direct reaction of the tantalum (powder 99,98 % Alfa Aesar) and antimony (powder 99,999 % Alfa Aesar) at 800 • C in a evacuated silica tube during 10 days.Starting from this microcrystalline powder, TaSb 2 crystallized by the chemical transport reaction in a temperature gradient from 1000 • C (source) to 950 • C (sink), and a transport additive concentration of 10 mg/cm 3 iodine (99,998% Alfa Aesar).The crystallization experiment was carried out in a horizontally arranged two-zone furnace which was tilted by approximately 10 • .The typical size of crystal obtained is 3 × 3 × 1 mm.The quality of the sample was checked by x-ray diffraction measurement.The local stoichiometry was investigated by energy dispersive x-ray (EDX) analysis (see Appendix A).Both data revealed that the synthesized sample is phase pure and of high homogeneity.The TaSb 2 phase was identified half a century ago [41] and its structure assigned to the NbSb 2 type [42].The roughly atomic coordinates were established from the evaluation of x-ray powder Guinier photographs [43].Although a detailed description of the crystal structure of TaSb 2 was therefore not yet available, our investigation using x-ray single crystal diffraction have refined detailed crystal structure (see Appendix A).Hereafter, we will use the refined crystal parameters to analyze NMR  data and band structure calculations.

III. NQR AND NMR TECHNIQUES
Our target nuclei in TaSb 2 are 121/123 Sb and 181 Ta and those nuclear parameters are tabulated in the Table I.The most prominent feature among those parameters is that 181 Ta has an extraordinary large nuclear quadrupole moment.This immediately suggests that 181 Ta-NQR occurs at a relatively high frequency.The NQR experiments were carried out on powder (crashed single crystals) sample using a conventional pulsed NMR/NQR spectrometer (Tecmag Apollo).The NQR spectra were taken by the frequency-sweep method where the real part of the spin-echo signal was accumulated as a function of frequency.In order to assign the observed NQR lines to nuclear isotopes and different sites, and to determine the principal axes of the EFG tensor, the weak field NMR (WFNMR) measurements were also carried out using an oriented single crystal, where the resonances were obtained under a weak field (less than 0.05 T) applied to the NQR lines.When a weak field applies along the main principal axis of EFG, the NQR lines splits into two lines.We have also measured the angular dependence of the WFNMR lines from where the principal axes of EFG for two Sb sites and one Ta site were determined.The temperature dependence of nuclear spin-lattice relaxation rate (1/T 1 ) has been measured for the NQR lines by the inversion recovery method.The T 1 values are obtained by fitting the recovery of nuclear magnetization, M (t), measured by the spin-echo amplitude after an application of inversion pulse, to the proper recovery function (refer Eqs.(B1) and (B2) in Appendix B).

IV. DFT BAND-STRUCTURE CALCULATIONS
First-principles calculations were performed using the full-potential code FPLO version 21, employing a localorbital basis set [44].To account for the sizable spin-orbit coupling in TaSb 2 , we use the full-relativistic treatment.The local density approximation (LDA) parametrized by Perdew and Wang [45] was employed for the exchangecorrelation potential.Unless otherwise mentioned, for the structural input we used the data obtained by our x-ray single crystal diffraction (see Appendix A).
For microscopic insights into the electronic structure of TaSb 2 , we resort to the nonmagnetic DOS depicted in Fig. 2 (a).The valence band splits into a high-energy manifold dominated by Sb 5p states (between −12.5 and −7.5 eV) and the low-energy manifold (starting around −6.5 eV and ranging up to the Fermi energy) to which also Ta 5d contributes prominently.The valence and the polarization bands are bridged by a steep valley with an intricate structure.Since this exact shape of this valley plays a crucial role in the transport properties of TaSb 2 , we resolve it in the inset of Fig. 2 (b).The most prominent feature in this energy range is the dip centered around 0.1 eV above the Fermi energy.The precise structure of the DOS is sensitive to the lattice constants and atomic positions: If we use the optimized TaSb 2 structural from the Materials project (mp-11697 [46]), this dip becomes sharper and moves closer to the Fermi level.Moreover, the difference between scalar-relativistic and full relativistic calculations reveals a profound impact of the spin-orbit coupling (SOC) on the electronic structure in this energy range.Interestingly, if we consider the full valence band, this difference becomes barely visible (Fig. 2, (b), main panel).
It has been suggested that a magnetic field may stabilize a Weyl point in TaSb 2 [32].While we can not exclude such scenario, the band structure plot reveals that highest-lying occupied band and the lowest-lying unoccupied band are separated by a finite gap (Fig. 2 (c)).Since the magnetic energy scale is roughly set by Bohr magneton which amounts to approximately 5.8×10 −5 eV/T, Comparison of DOS calculated in the full relativistic ("with SOC") and scalar-relativistic ("no SOC") mode for two structures: the experimental structure obtained from our x-ray single crystal diffraction (see Appendix A) and the optimized structure from the Materials project (mp-11697 [46]).The inset shows a blow-up of the same data.(c) Band structure for the experimental crystal structure.In all plots, the Fermi level (dashed line) is at zero energy.
closing a gap of 0.1 eV would require an unrealistically large magnetic field.

A. NQR spectra and site assignment
Searching for the NQR signal in TaSb 2 , we consider the splitting of nuclear energy levels due to the quadrupole interaction, which is expressed by the following Hamiltonian, where the asymmetric parameter, V xx , V yy and V zz are the EFG at nuclear sites along the x-, y-, and z-axis, respectively.
Here, we note that the z-axis is the maximum principal axis of the EFG tensor, V zz , and nuclear spins are quantized along V zz at zero field.The quadrupole-split nuclear energy levels E m and the resulting transition frequencies can easily be calculated numerically by diagonalizing Eq. ( 1).For η = 0, the energy levels can be expressed as .
Then the NQR frequency can be written as ν N QR = ν Q (2|m|+1)/2, because the NQR occurs for the transition between two levels |m| and |m + 1|.Therefore, for η = 0, two and three NQR lines with equal spacings are expected for I = 5/2 and I = 7/2, respectively.For η̸ =0, those NQR lines will be shifted with un-equal spacing [47].Hence, a total of thirteen NQR lines should be observed for 121 Sb(1), 123 Sb(1), 121 Sb(2), 123 Sb(2) and 181 Ta sites in TaSb 2 .In Fig. 3 (a), (b) and (c), we show the observed NQR spectra at 5 K.The expected thirteen-NQR lines have clearly been observed in the frequency range of 10∼260 MHz.The peak frequencies for each line have been obtained by fitting each line to a Gaussian function.
The following procedures have been taken for the site assignment, namely how we can associate each line to the particular sites and nuclear isotopes.First, we consider the nuclear quadrupole moment (Q) for each nucleus to assign which site the spectrum originated from.The Q of 181 Ta ( 181 Q = 3.170×10 −28 m 2 ) is approximately five times larger than that of 121/123 Sb ( 121 Q = 0.543×10 −28 and 123 Q = 0.692×10 −28 m 2 ).Moreover, the 121 / 123 Sb NQR lines at the same site are expected to be observed in a close frequency range since the ratio of 121 ν Q to 123 ν Q at the same site is 1.47 from Eq. ( 2).Here, for the sake of clarifying, the observed NQR lines are divided into three groups according to frequency ranges of observed NQR lines and assigned as G(1) (15∼40 MHz) for 121/123 Sb, G(2) (40∼75 MHz) for 121/123 Sb, G(3) (110∼255 MHz) for Ta.
Next, we estimate the ν Q from the combinations of all spectra within each group.The estimated ν Q for 121 Sb and 123 Sb at each site should correspond to each other when the site-assignment is correct.Here, we also consider the asymmetry parameter η because the peak frequencies are unequally spaced for any combinations of spectra.In order to estimate the values of ν Q and η from the observed peak frequencies at 5 K, a least-squares analysis was performed using theoretical quadrupole interaction obtained from the exact diagonalization.The values of ν Q and η have been estimated from all combinations of NQR lines within each group.We found the combinations of NQR lines in which the ν Q and η for 121 Sb correspond to that for 123 Sb, as shown in Table II.Therefore we conclude that G(1), G(2) and G(3) are associated with respectively Sb(1), Sb(2) and Ta sites.
The quadrupolar frequency ν Q and the asymmetry parameter η can be obtained from the EFG tensors at nuclear sites defined as the second partial derivative of the electrostatic potential v(r) at the position of the nucleus: where δ ij and ∇ 2 are the Kronecker delta and the Laplace operator, respectively.In the DFT calculation, we choose Ta (4f /5s5p6s5d6p5f ) and Sb (4s4p4d/5s5d5p4f ) semicore/valence states as a basis set.The inclusion of higher lying states into the basis is crucial for an accurate estimation of the EFG tensor [33].The calculated ν Q and η are provided in  a relative measure, and its distribution law differs from that of individual EFG components.Assuming that the latter are normally distributed, the distribution of η can be approximated as normal, even though with the ∼1.5 times larger standard deviation.This difference has to be taken into account for a proper comparison.

B. Weak Field NMR and determination of the principal axis
To experimentally determine the principal axes of the EFG tensor, a weak magnetic field (H) was applied to the single crystal.Here, we can treat the Zeeman term H z = −γ N H(I x sin θ cos ϕ + I z cos θ) as the perturbation of H Q , where θ and ϕ are the angle between a magnetic field and the principal axis of V zz in the x − z plane and x − y plane, respectively, and γ N is the bare nuclear gyromagnetic ratio.Then one NQR line splits into two lines (f WFNMR −m and f WFNMR +m ).In the case of transitions between ±1/2 and ±3/2, transitions between m and m±2 also occur, so that exceptionally one NQR line splits into four lines [47].Therefore, generally six and eight WFNMR lines are expected for I = 5/2 and 7/2, respectively (see detail in Appendix C).We define the split- where γ ′ is an apparent nuclei gyromagnetic ratio and the ∆f exhibits an angular dependence (see detail in Appendix C).The typical example of the WFNMR spectra at 121 Sb(1) (ν N QR = 33.46MHz) is shown in Fig. 4 (a).The clear splitting of the spectrum has been observed.The peak frequencies for each magnetic field have been obtained by fitting each spectrum to two Gaussian functions.The line width of the spectra does not vary with the magnetic field and no additional peaks were observed.This fact indicates that magnetism or magnetic impurities are absent.As shown in Fig. 4 (b), the field dependence of peak frequencies is linear, and the slope is 11.875 MHz/T at θ ′ = 50 • , where θ ′ is an angular between the magnetic field and the axis perpendicular to the c-axis in the acplane.The slope greater than γ = 10.189 is due to the effect of finite η.The principal axis of V zz and the associated isotope can be assigned by examining the angular dependence of ∆f /2H(∆f /2H(θ ′ )).We have measured the angular dependence of WFNMR at the highest NQR frequency lines for G(1) and G(2) and the f 2 line for G(3).For example, the angular dependence of WFNMR spectra and peak frequencies for 121 Sb(1) are shown in Fig. 5 (a) and (b), respectively.Also, the angular dependence of ∆f /2H is shown in Fig. 5 (c).We found that the principal axis of V zz for Sb(1) is pointing θ ′ = 50 • , ϕ ′ = 0 and η = 0.42, where ϕ ′ is the angle between the principal axis of V zz and the ac-plane.Note that ϕ ′ =0 means that V zz being within the ac-plane.
Similar measurements were made at 121 Sb(2) and 181 Ta, then we have found that the principal axis of V zz pointing θ ′ = 107 • and 125 • , ϕ ′ = 0 and 0, η = 0.74 and 0.32, respectively.The obtained η values are consistent with the experimental values obtained from the satellite structure shown in Table II.From the DFT calculation, we have obtained the direction of the principal axis of V zz as θ ′ = 69 • , 122 • , and 135 • for Sb(1), Sb(2) and Ta sites, respectively.Therefore, the experimentally obtained direction of principal axis of V zz is in good agreement with In order to extract the temperature dependence of the quadrupolar parameters, ν Q and η, the temperature dependence of the NQR frequencies for all lines have been measured up to 250 K.The results are shown in Fig. 6 (a), (c), and (e).The ν Q (T ) at each site follows an empirical ν Q (0)(1 − AT 3/2 ) law below 50 K.This is a typical temperature dependence reflecting the temperature dependence of thermal expansion [48].From the least-square fit of ν Q (T ) to ν Q (0)(1 − AT 3/2 ), we obtained the ν Q (0) at each site, as shown in Fig. 6 (b), (d) and (f).To discuss site dependence of EFG, the fractional decrease of V ii (T ) (i = x, y, z), ∆V ii (T ) = (V ii (0) − V ii (T ))/V ii (0) are shown in Fig. 7 (a),(b) and (c).The deviation in ∆V zz (T ) among sites is less than 0.06% over the whole temperature range, and there is hardly seen any site dependence on ∆V zz (T ).In contrast, η(T) show different behavior depending on the site: at the Sb(1) and Ta sites, the η(T) decreases with increasing temperature, whereas at the Sb(2) site, η(T ) increases (see Fig. 6 (d)).
The site-differentiation might be due to the anisotropy of the thermal expansion of lattice parameters.Here we extract V xx and V yy (see detail in Appendix D) to discuss η(T) in more detail at each site.Fig. 7   tively.First, as a characteristic behavior, both ∆V xx and ∆V yy depend on temperature at the Sb(1), whereas ∆V yy almost no temperature dependence was observed at the Sb(2).The temperature independent of ∆V yy is the main source of the unusual increase of η(T ).Based on the DFT calculation, only ∆V yy for Sb(2) is pointing to a-axis, as shown in Fig. 7 (e).This result indicates that the lattice thermal expansion of the a-axis might be smaller than the others.Next, the ∆V yy for Sb(1) and Ta and ∆V xx for Sb(2), pointing to the b-axis, are increasing, that indicates thermal expansion induces in the b-axis direction.The temperature dependence of EFG tensors pointing in the ac-plane demonstrates that ∆V zz shows an increase with increasing temperature at all sites, whereas ∆V xx shows a decrease at Sb(1) and Ta.If one considers thermal expansion as the sole cause of these temperature dependencies, a contradiction arises because ∆V zz indicates an expansion and V xx indicates a shrinkage.For a deeper insight into the mechanism of thermal expansion, low-temperature diffraction experiments and dilatometry studies are highly recommended.Before discussing the temperature dependence of 1/T 1 based on the band structure, it is necessary to clarify that the relaxation process is governed by the magnetic excitations, rather than the quadrupolar process.One way of doing this is the careful analysis of the recovery of the nuclear magnetization from the saturated state to the thermally equilibrium state.From the recovery of nuclear magnetization, it was confirmed that the relaxation process is governed by magnetic excitation (See detail in Appendix B).Furthermore, the experimentally obtained T 1 ratio between 121 Sb and 123 Sb, 121 T 1 / 123 T 1 , is a good measure of whether the relaxation process is governed by magnetic or quadrupolar, because the former case the ratio should be ( 123 γ N / 121 γ N ) 2 = 0.29 while the latter case it should be ( 123 Q/ 121 Q) 2 = 1.62.We have obtained the experimental ratio of 0.30 and 0.29 at Sb(1) and Sb(2), respectively.These values agree well with the ( 123 γ N / 121 γ N ) 2 .From the above two evidences, we are strongly confident that the relaxation process in TaSb 2 is to be discussed from magnetic excitations.
Putting the experimentally obtained temperature de-pendence of Q n and K n due to those of η(T ) (see Fig.  8 where the 1/T 1 T are divided by the square of the γ N to eliminate nuclear specific parameter.Although the general feature has been seen in many semi-metallic materials, one can immediately observe that there exists a characteristic temperature, T * ≈ 30 K, where the relaxation process has a crossover from a high temperature activation type behavior which is presumably associated with the excitations to a higher unoccupied state to a low temperature modified Korringa excitations.Among topological materials, especially in the Dirac/Weyl fermion system with linear band dispersions, a power low behavior 1/T 1 T ∼ T n has been observed at high temperatures [33,38].The power low behavior originated from a Fermi-contact and/or an orbital contribution of hyperfine interaction.When the Fermi-contact contribution is dominant, T 4 dependence is predicted due to the linear dispersion with the point node.While when the orbital contribution is dominant, an unusual temperature dependence of hyperfine interaction leads to a T 2 dependence in 1/T 1 T [49,50].However, neither case can explain the activation-type behavior.This result is consistent with the absence of Weyl points at zero magnetic field in TaSb 2 from band structure calculations. Quite generally, 1/T 1 T can be expressed by using the wave vector (q) and frequency (ω), dependent magnetic susceptibility, χ(q, ω), characterizing the magnetic excitations in a system as [51,52], where χ ′′ ⊥ (q, ω) is the transverse component of imaginary part of χ(q, ω), A q is the q-dependent hyperfine coupling constant and ω N is the frequency for the NQR.Since at the moment we do not have a plausible microscopic theory to calculate the χ(q, ω) in multiband systems like TaSb 2 , we have adopted the theoretical 1/T 1 T for non-interacting itinerant electrons based on the band structure calculation with random phase approximation (RPA).For such a system 1/T 1 T is expressed using the density of state near the Fermi level as [53], where f (E) is a Fermi distribution function, D(E) is the energy dependent density of state and µ c (T ) is the temperature dependent chemical potential.Were there no change of A hf with temperature, the calculation of 1/T 1 T is straightforward from calculated D(E) (see Fig. 2 (a)).At all sites, the calculated 1/T 1 T shows a constant value without increase.Namely, neither the low temperature up-turn nor the high temperature activation type behaviors are reproduced by the calculated 1/T 1 T based on the DOS.The absence of sharp DOS features in the immediate vicinity of the Fermi level excludes the possibility of sizable shifts of the µ c (and hence the DOS at µ c ) at low temperatures.Thus, the nature of the experimentally observed upturn in 1/T 1 T remains elusive in the band structure.One possible explanation is that the DFT fails to describe the electronic structure of TaSb 2 accurately.While deficiencies of DFT can not be excluded especially for highly covalent systems, we have two arguments that render such a scenario unlikely.First, our DFT calculations accurately describe the EFG tensors.Second, our results are in agreement with the linear coefficient γ = 1.2 mJ•mol −1 •K −2 estimated from the Sommerfeld expansion of the specific heat at low temperatures (see Appendix E).Indeed, γ can be estimated directly from the DOS at the Fermi energy D ε F (in J −1 per formula unit) where k B is the Boltzmann constant (in J/K), N A is the Avogadro number (in mol −1 ), Z = 2 is the number of formula units per cell, and V uc is the unit cell volume (in m 3 ).In the full relativistic calculations, the resulting γ DFT values amount to 1.00±0.05mJ•mol −1 •K −2 .We also simulated the very small possible deviation of the sample stoichiometry from the ideal 1:2 ratio (see Appendix A) applying the virtual crystal approximation and found no significant changes in the DOS near the Fermi level.The agreement between the calculated and the experimental values also not only supports our DFT analysis, but also indicates that band renormalization in TaSb 2 is small.The discrepancy between the experimental 1/T 1 T (T ) and those from the band structure calculation demonstrate that TaSb 2 is not a simple semi-metal, and the magnetic excitation has to be treated beyond the simple Fermi liquid scenario.Following a number of previous NMR studies like SmB 6 [54], PuB 4 [55] and FeGa 3 [56], we assume an existence of rather arbitral in-gap state which is invisible in calculated band structure to analyze the 1/T 1 T .Here, D(E) exhibits a symmetric band gap δ centered at E F shown in inset of Fig. 8 (b).The origin of the narrow band in-gap state is not clear but may be associated with the Anderson localization or impurity states.Assuming a simple in-gap state with narrow bands which is invisible in the calculated band structure, the activation-type temperature dependence of 1/T 1 T (T ) is discussed by a phenomenological way.The data are then fitted to the following empirical equation, including the upturn at low temperatures, The first term describes a fluctuation of the localized electron moments within the narrow band at low temperatures, and the second term describes thermal excitation of the quasiparticles.The solid lines in Fig. 8 (a) and (b) indicate such fit of the 1/T 1 T (T ) to Eq. 7 with δ = 13 meV(150K), 6 meV(70K) and 23 meV(260 K) for Sb(1), Sb(2) and Ta sites are obtained, respectively.The obtained different activation energies at each site might indicate the presence of site dependent gap.As indi-cated by the site dependence of the gap size, the increase of 1/T 1 T with temperature at the Ta site is larger than at the Sb site.Such differences in site-specific behavior have also been observed in Weyl semimetal TaAs [57].The origin of the site dependence is unknown, but a theoretical study is warranted.
The power law coefficient β in the first term is associated with the strength of exchange coupling between lower occupied state where β=1 and β=0 correspond to purely localized state and delocalized state, respectively.The fit value of β∼0.5, averaged for all sits in TaSb 2 , as well as for TaAs [57], may indicate that the assumed in-gap state is rather spatially extended so that the exchange narrowed theory which predicts β = 1 may not be applicable.This explanation is on the assumption of in-gap states associated with the Anderson localized or impurity state as a phenomenological description, and the following possibilities for the origin of the upturn can also be mentioned.i) Contribution from the mobility edge near the gapped band [58], ii) Fluctuations in the Kondo state (impurity induced Kondo hole) [59] iii) Spin diffusion to the magnetic impurity in topological insulator [60].Nevertheless, the idea that the nuclear spin diffusion to dilute impurity moments is discussed in YbB 12 [60] seems to be one of the plausible interpretations.Likewise, the similar mechanism from bulk to surface metallic state in topological materials is worth investigating from theoretical and experimental viewpoints.

VI. CONCLUDING REMARKS
The presented study is a combination of experimental nuclear quadrupole resonance (NQR) and theoretical calculations using crystal structure obtained from our x-ray single crystal diffraction.We have reported characteristic static and dynamical properties of observed thirteen NQR lines associated with 121 Sb, 123 Sb and 181 Ta in TaSb 2 .All 121/123 Sb and 181 Ta NQR lines are successfully assigned to all inequivalent sites and isotopes by combined analysis of the DFT calculation and the WFNMR measurement.The following characteristic features are to be noted.
(I) From the data fitting of the temperature dependence of NQR lines the zero-temperature quadrupole coupling constant ν Q and asymmetry parameter η are extracted, and those values are in agreement with the DFT calculations within ±16%.The temperature dependence of ν Q (T ) and η(T ) has been measured up to 250 K, and unusual η(T ) was observed for the Sb(2) site.The angular dependence of WFNMR frequencies for all sites enabled to determine the direction of the EFG principal axes, V xx and V yy and V zz with respect to the monoclinic crystal axes.The temperature dependencies of the V xx (T ) and V yy (T ) and V zz (T ) have shown a unique characteristic where we found the V yy (T ) of Sb(2) site has almost temperature independent.Since the V yy of Sb( 2) is pointing exactly to the a-axis, the origin of the unexpected η(T ) may be attributed to the small lattice expansion of the a-axis.
(II) The nuclear relaxation process measured in the NQR at all the sites is proved experimentally by the magnetic excitations, not the quadrupolar.The temperature dependence of relaxation rate, 1/T 1 T (T ), has quite characteristic behavior similar to the many semimetal materials.That is the high temperature activation type process crosses over to the low temperature excess relaxation process, making an upturn in 1/T 1 T (T ) around 30-70 K.However, neither high nor low temperature behaviors could be explained by the simple Fermi liquid using the DFT density of states at Fermi level, even the temperature dependence of the chemical potential is included.Followed by the traditional procedure, the activation type behavior of 1/T 1 T (T ) was analyzed by assuming simple in gap state could reproduce high temperature 1/T 1 T (T ).This energy scale of the activation is similar to other topological materials.We did not observe any power-law behavior in 1/T 1 T (T ) characteristic to Weyl or Dirac fermion excitations.
Our DFT-supported NQR/NMR study with the local probes Ta and Sb provide with a valuable contribution to understanding of the temperature dependent local charge distribution, band structure and magnetic excitations in the topological semimetal TaSb 2 .
where c 0 is the degree of inversion, and Q n (n = 1,2,3) and K n (n = 1,2,3) are the parameters depending on the nuclear spin I and η.All parameters are calculated in Ref. [61] for a given I and η.The typical examples are shown in Fig. B1, where the experimental recovery curves for the f 2 lines in 121 Sb(1), 121 Sb(2) and 181 Ta NQR lines are shown with the best fit curve to Eqs. (B1) and (B2) using K n and Q n shown in Table B1.The obtained perfect fit provides us with the relaxation process is governed by the magnetic fluctuations.We obtain nuclear energy levels, E m , by the Zeeman effect resulting from the exact diagonalization of matrix which consists of Eqs.(C1), (C2) and (C3).When η = 0 and θ = 0 (H ∥ V zz ), WFNMR occurs for the transition between two levels m and m±1 (|m|>3/2), and the resonance frequency can be expressed as, f W F N M R ±m = ν N QR ± γ N H.
Following Eqs.(C1), (C2) and (C3), we know that a measurement of the angular dependence of ∆f /2H makes it possible to determine the principal axis direction of V zz (θ and ϕ) and η.We have calculated the angular dependence of ∆f /2H using the exact diagonalization of Eqs.(C1), (C2) and (C3).Here, we assume that V zz is along θ = 0 • , ϕ = 0 • and γ = 1 MHz/T.The maximum of ∆f /2H increases with η, as shown in Fig. C1 (a).And the ∆f /2H at 90 • also changes with η.On the other hand, Fig. C1 (b) shows that ϕ has no effect on the maximum value of ∆f /2H, but the 90 • component decreases with ϕ.Therefore, the direction of V zz and η can be obtained by analyzing the experimental results of the angular dependence of ∆f /2H using a least-squares analysis with η and ϕ as parameters.

Appendix D: Temperature dependence of EFG tensor
To examine the EFG tensor in detail, V xx , V yy and V zz were obtained by decomposing the temperature dependence of ν Q (T ) and η(T ).The V zz (T ) was determined based on Eq. 2 in the main text, and the sign was determined by the DFT calculation (see in Table II).Since the EFG tensor is defined as |V zz | > |V yy | > |V xx | and V zz +V yy +V xx = 0, V xx and V yy can be written as V xx = V zz (η-1)/2 and V yy = -V zz (η+1)/2.Thus we determined V xx (T ), V yy (T ), and V zz (T ) for each site are shown in Fig. D1(a∼ i).

Appendix E: Specific heat
The specific heat of a TaSb 2 single crystal was measured in the temperature range between 2 K and 300 K using Quantum Design PPMS system (Fig. E1).At high temperatures, the Dulong Petit law was proven.At low temperatures, the data can be described with a linear (electron) and a cubic (phonon) dependence.It is clear that i) there is no increase in C/T which means the increase in 1/T 1 T at low temperatures may not be electron origin and ii) the determined value for the linear part is 1.2 mJ/molK 2 which is within the range of what is estimated from the band structure calculations (Fig. 2  the main text).

4 FIG. 2 .
FIG. 2. Electronic structure of TaSb2 in nonmagnetic LDA calculations.(a) Total and site-resolved density of states (DOS) in a full-relativistic non-mangetic calculation.(b)Comparison of DOS calculated in the full relativistic ("with SOC") and scalar-relativistic ("no SOC") mode for two structures: the experimental structure obtained from our x-ray single crystal diffraction (see Appendix A) and the optimized structure from the Materials project (mp-11697[46]).The inset shows a blow-up of the same data.(c) Band structure for the experimental crystal structure.In all plots, the Fermi level (dashed line) is at zero energy.

FIG. 4 .
FIG. 4. Typical example of the field dependence of WFNMR spectra (a) and peak frequencies (b) for 121 Sb at 33.46 MHz NQR line.The weak fields were applied along θ ′ = 50 • .The slope is estimated as 11.875 MHz/T (see main text).Note that the bare gyromagnetic ratio of 121 Sb is 10.189 MHz/T.

FIG. 5 .
FIG. 5. Typical example of the angular dependence of 121 Sb(1) WFNMR frequency.(a) FFT spectra taken at 33.46 MHz and 0.0082 T external field.Here a single crystal was rotated along the b-axis.(b) Angular dependence of peak frequency of the spectra.(c) Angular dependence of ∆f /2H extracted from exact diagonalization.Solid lines are data fits to theoretical HQ + Hz from exact diagonalization where νQ = 17.5 MHz and η = 0.42 were obtained.
(a) and (b) show the ∆V xx (T ) and ∆V yy (T) for each site, respec-
FIG. A2.X-ray powder diffraction recorded in Guinier geometry (black line).The difference between measured and calculated intensities is represented as a red curve.Blue vertical bars indicate reflection positions of the refined model.
FIG. B1.Recovery of nuclear magnetization M (t) measured by the spin-echo amplitude after an inversion pulse for 121 Sb ((a) and (b)) and 181 Ta ((c)) NQR lines at 5K.The black solid lines are data fits to Eqs. (B1) and (B2) for 121 Sb and 181 Ta, respectively.
FIG. E1.(a) Specific heat as a function of temperature.DP denotes the Dulong Petit value.The curved dotted line indicates a description of the data below 10 K with electron and phonon contribution.(b) Plot of the specific heat C/T versus T 2 .The straight line indicates a description with electron and phonon contribution.The intercept at T = 0 indicates an electronic contribution to the specific heat.

TABLE I .
NMR/NQR related nuclear parameters of 121 Sb, 123 Sb, and 181 Ta

TABLE II .
Experimental results and calculated EFG tensor Vzz, quadrupole coupling νQ and asymmetry parameter η for each site in TaSb2.Sb(1) and Sb(2) are inequivalent Sb sites and the labeled one is consistent with the one named for the crystal structure.

TABLE A2 .
Atomic coordinates and equivalent displacement parameters (in Å 2 ) in the crystal structure of TaSb2.