Introduction

Since a number of heavy half-Heusler ternary compounds were proposed as a new class of three-dimensional topological insulators with a tunable nontrivial band structure as a result of strong spin-orbit coupling (SOC)1,2,3,4, this kind of compounds have attracted considerable attentions as multifunctional quantum materials5,6,7. Their nontrivial bulk band structure with relativistic-like character has proven to be closely linked to their strong SOC8,9,10, as is inferred from their unique transport properties such as extremely high carrier mobility, large linear magneto-resistance and pronounced weak anti-localization effect11,12,13,14,15,16,17. Similar to the well-known topologically nontrivial HgTe compound18,19, the inversion of an s-like Γ6 band in heavy atom containing half-Heusler compounds, identified by numerical computations, is interesting and claimed to result from relativistic effects20. Meanwhile, the SOC effect, which is purely relativistic in origin10,21, is expected to split the degenerate pxyz-Γ5 states (primarily being t2g and p orbitals) into quartet pxy-Γ8 ( j = 3/2, denoted as p3/2) close to the Fermi level and doublet pz-Γ7 ( j = 1/2, denoted as p1/2) states below the Fermi level, as illustrated in Fig. 1(a–c). The experimental determination of their topologically nontrivial nature is of fundamental importance for setting the foundation of their potential applications. However, experimental verification remains limited to comparing band calculations with conventional angle-resolved photoemission spectroscopy (ARPES)7,22. Unfortunately, it is likely that technical challenges using this type of surface-sensitive method have precluded success in revealing the topologically nontrivial band structure nature of unstrained bulk half-Heusler compounds22.

Figure 1
figure 1

Crystal structure and bulk electronic states of Half-Heusler R-M-Bi.

Schematic diagrams of (a) the crystal structure, (b) bulk electronic band states without SOC (denoted as W/S) and (c) with SOC, respectively. For the band structures, SOC splits the pxyz-Γ5 state into pxy-Γ8 (j = 3/2, denoted as p3/2) and pz-Γ7 (j = 1/2, denoted as p1/2) states. The SOC parameter is defined as ESOC = EΓ8 − EΓ7, where a larger ESOC shows a stronger SOC strength in system. An illustration of SOC effects on NMR chemical shifts (d–f). The SOC-split p1/2 state imposes an additional magnetic shielding to the nucleus through hyperfine interactions. The direction of shielding tensor from SOC effects can be different, depending on the nature of exchange interactions.

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Because relativistic effects influence nuclear magnetic shielding, relativistic modifications of the bulk band structure should, therefore, be detectable by locally susceptible sensitive methods such as nuclear magnetic resonance (NMR)23,24. For example, the probability density of relativistic p1/2 states at the nucleus is not zero25, therefore, we can expect additional contributions to the nuclear magnetic shieldings in these bulk half-Heusler compounds [see Fig. 1(d–f)]. Magnetic shieldings will not cancel out due to the different occupancy of p1/2 tuned by relativity among the members in this family of compounds26. Moreover, nuclear magnetic shielding is sensitive to the change of symmetry of the conduction bands owing to the inversion of the s-like Γ6 band. Although relativistic effects including SOC on NMR chemical shifts and nuclear spin-spin interactions on small molecules and complexes21,27,28 and on the surface states of Bi2Te324 have already been widely acknowledged and discussed, it is highly desirable for systematic investigation of the relativistic effects on the NMR spectra in heavy half-Heusler intermetallics29,30,31. In addition, the underlying mechanism of the correlation between NMR parameters and topologically nontrivial nature remains inconclusive.

In this work, we demonstrate the relativistic effects, including SOC on bulk band structures in topological half-Heusler R-M-Bi (R = Sc, Y, Lu, M = Ni, Pd, Pt) compounds, by combining bulk 209Bi NMR spectroscopy with a state-of-the-art first-principles calculation. Here, 209Bi (spin I = 9/2, 100% natural abundance) NMR is selected as an atomic probe because Bi is a common heavy component in these RMBi half-Heusler compounds. In addition, 209Bi NMR is one of the most sensitive atomic probes to changes in chemical shieldings at Bi sites in a cubic structure where quadruple interactions can be neglected. We find that 209Bi NMR shift tensors scale with the relativity represented by average nuclear charge <Z>, as well as the band topology characterized by EBIS (defined as energy difference between Γ6 and Γ8 bands, EBIS = EΓ6 − EΓ8) or ESOC (ESOC = EΓ8 − EΓ7) in these half-Heusler compounds. Evidence presented here for relativistic effects on bulk band structures probed by 209Bi NMR provides an important insight into the mechanism of band inversion and its role on transport properties in these topologically nontrivial half-Heusler compounds.

Results

Crystal growth and characterization

Single R-M-Bi crystals were grown out of Bi flux and characterized by X-ray diffraction (XRD) with Cu Kα radiation on a Rigaku x-ray diffractometer. Details on growth procedures and XRD characterizations can be found in the supplemental materials. These nonmagnetic compounds crystallize in a cubic MgAgAs structure with a space group number of 216 (F-43m), which can be viewed as three face-centered cubic sublattices positioned at (0, 0, 0), (l/4, 1/4, 1/4) and (3/4, 3/4, 3/4) along the body diagonal. 209Bi NMR spectra of powdered single crystals were obtained with a Bruker Avance III 400 spectrometer in a magnetic field of ~9.39 Tesla. A Hahn-echo pulse sequence was used to record echo intensity with the last delay time around 300 ms because of the short spin-lattice relaxation time (~60 ms). As determined from nutation experiments, the measured typical 90 degree time (tπ/2) is about 5 μs for the samples, close to that of aqueous Bi(NO3)3 at the same r.f. condition. 209Bi isotropic shifts were referenced to a 1M Bi(NO3)3·5H2O aqueous nitrate solution (8.4 ppm at 298 K).

Analysis of composition dependence of NMR spectra

Figure 2(a) presents typical static 209Bi (spin I = 9/2) NMR spectra measured at room temperature for powdered single crystals of isoelectronic LuPd1-xPtxBi (x = 0, 0.2, 0.9 and 1). Only 209Bi central lines can be observed, as expected for the cubic symmetry. The central linewidth defined as full line width at half magnitude (FWHM) is ~20 kHz for LuPtBi, which is much larger than that estimated for pure dipolar broadening (~1 kHz), indicating the existence of other contributions to the broadening such as second-order quadrupole, chemical-shift anisotropy or site distributions. Potentially, the atomic disorder will break the local cubic symmetry at Bi sites and lead to site distributions. Since the measured quadrupole coupling constants are very small, it is reasonable to neglect the contributions of 209Bi central line broadening and anisotropic shifts from second-order quadrupole interactions. The observed nearly symmetrical central line shape for LuPdBi within which the peak locates at the center of gravity, we can thus define 209Bi isotropic shifts as the peak position of the central lines. Meanwhile, the central line shape of LuPtBi is asymmetric and, therefore, the positions of the isotropic resonances are simulated by shift anisotropy. In addition, site distributions make no contributions to isotropic shifts.

Figure 2
figure 2

209Bi NMR spectra in isoelectronic alloys LuPd1-xPtxBi (x = 0, 0.2, 0.9, 1).

(a) Typical 209Bi powder spectra of half-Heusler alloys LuPdBi, LuPd0.8Pt0.2Bi, LuPd0.1Pt0.9Bi and LuPtBi. All spectra are normalized with respect to peak intensity. The peak positions are referenced to Bi(NO3)2 in aqueous solution. (b) The measured NMR 209Bi isotropic shifts and the calculated SOC parameter ESOC for the half-Heusler system LuPd1-xPtxBi.

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Similar to the well-studied Bi1-xSbx topological insulators, the SOC strength in LuPd1-xPtxBi can be tuned within a certain range of x32. Figure 2(b) reveals that ESOC (=EΓ8 − EΓ7) derived from band structure calculations, as an indicator of SOC strength, increases in magnitude with Pt alloying. Details of the calculations and analysis are described in the supplemental materials. Simultaneously, 209Bi NMR resonant frequency moved toward upfield upon Pt alloying. The measured 209Bi isotropic shift is about −6300 ppm for LuPtBi and −4300 ppm for LuPdBi. Similar values for the two compounds measured at the same temperature but at a different magnetic field have been reported31, indicating that these values are independent of magnetic field. Upon Pt alloying in LuPd1-xPtxBi, the 209Bi isotropic shifts change from −4830 ppm for LuPd0.8Pt0.2Bi, to −6059 ppm for LuPd0.1Pt0.9Bi. Note that Fig. 2(b) shows that the calculated ESOC and measured 209Bi shifts are strongly correlated with each other, where Pt concentration is an implicit parameter. This result indicates that the SOC parameter is coupled with 209Bi NMR isotropic shifts in LuPd1-xPtxBi series.

Scaling of 209Bi NMR isotropic shifts

Since the relativistic modifications of band structures have a similar mechanism in these R-M-Bi compounds1,2,10, it would be interesting to link 209Bi NMR shifts with EBIS or ESOC if possible. For a convenient comparison, Table 1 lists the measured 209Bi isotropic shifts, the calculated EBIS, ESOC values and their topological status. Figure 3(a) shows that rough correlations can be built between 209Bi isotropic shifts and average SOC strength in terms of ESOC if the R-M-Bi compounds are treated as two subgroups for consideration, where five members (ScNiBi, YNiBi, LuNiBi, LuPdBi, LuPtBi) form one subgroup and four members (ScPdBi, YPdBi, ScPtBi, YPtBi) form the other. The average SOC strength can be also represented by average atomic number <Z> or nuclear charge. In general, the SOC constant21, is proportional to Z (nuclear charge number), where r refers to radial distance and other symbols have their normal meanings. Therefore, nuclear magnetic shieldings will scale with Z when relativistic effects are important. Figure 3(b) clearly shows that 209Bi isotropic shifts almost linearly follow <Z> for the two subgroups, where <Z> = (ZR + ZM + ZBi)/3 in R-M-Bi compounds. With <Z> increasing from ~40 to ~80, 209Bi shifts from about +3000 ppm to −6500 ppm. These results indicate that relativistic effects are relevant for understanding the mechanism of 209Bi NMR isotropic shifts in these half-Heusler compounds. The average SOC strength is believed to contribute to the value of EBIS1,20. In Fig. 3(c) we observe clear correlations between 209Bi isotropic shifts and EBIS in R-M-Bi compounds, especially the changing along subgroups labeled by a particular binary MBi = (PtBi, PdBi and NiBi).

Table 1 The measured values of lattice constants, NMR 209Bi isotropic shifts at room temperature, calculated band inversion strength EBIS (=EΓ6 − EΓ8), SOC parameter ESOC (=EΓ8 − EΓ7) and the topological status for half-Heusler R-M-Bi alloys.
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Figure 3
figure 3

Scaling of 209Bi isotropic shifts.

The measured NMR 209Bi isotropic shifts as a function of (a) the calculated SOC parameter ESOC, (b) the average nuclear charge <Z> (=(ZR + ZM + ZBi)/3), (c) the calculated band inversion strength EBIS and (d) the calculated NMR 209Bi isotropic shifts.

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To further reveal the relativistic effects on NMR shifts, a relativistic density functional approach to calculate spin-orbit shifts involving heavy 209Bi nuclei was performed at various relativistic levels. All calculations are implemented in WIEN2k code using the augment plane-wave DFT-NMR method33,34,35 and calculated band structures were similar to previous reports4. Previously, this method was used successfully to calculate NMR chemical shifts in small molecular systems or diamagnetic semiconductors35. To calculate 209Bi isotropic shifts in solids containing heavy atoms, relativistic SOC effects are invoked to account for the relativistic p1/2 electrons. Figure 3(d) shows the total calculated 209Bi chemical shieldings with SOC included. Fittings of the calculated chemical shieldings with respect to the experimental shifts yield the calculated 209Bi isotropic shifts, as practiced in ref. 35. As shown in Fig. S4, One can see that the calculated 209Bi shifts with SOC follow the measured ones more closely than the calculated ones with only scalar relativistic corrections. In other words, computations indicate that SOC plays a significant role in 209Bi NMR isotropic shifts in these heavy half-Heusler compounds.

In addition to isotropic shifts, traces of hidden relativistic effects on NMR can also be revealed through bonding properties and nuclear spin-spin interactions (up to several kHz) that are usually less than quadruple or dipole-dipole couplings in solids. As shown in Fig. 2(a), the central line shape of the 209Bi powder spectra of LuPtBi compound is asymmetric. Based on symmetry analysis, dipolar interactions do not account for the shift anisotropy in cubic crystals. This anisotropy is potentially an indicator of strong nuclear spin-spin interactions originating from the phase factor for relativistic p1/2 orbitals36. The relativistic increase of the shift anisotropy seems to be a general phenomenon that has also been found in small molecular systems21. Nuclear spin-spin coupling constants between 6th-row elements have been predicted to increase by almost one order of magnitude in comparison with their 5th-row counterparts, in line with the well-known relativistic bond contraction. For instance, nuclear spin-spin coupling constant can reach as high as 57 kHz for Pt-Tl21. Meanwhile, the central line width of a 209Bi NMR spectrum taken at room temperature is ~20 kHz. This line width is free of 2nd-order quadrupolar broadening and almost one order of magnitude larger than that from nuclear spin dipolar interactions (~1 kHz), as discussed earlier, indicative of the line broadening contribution from strong nuclear spin-spin coupling. Enhanced nuclear spin-spin coupling also explains the reduced degree of antisites in Lu-Pt-Bi, which is perfectly consistent with our experimental observations. LuPtBi is found to be stoichiometric and free of antisites, as indicated by XRD refinement. Within the lattice, M-Bi forms a covalent-type ZnS substructure7 from orbital hybridization and the lack of antisites suggests that the Lu-Pt-Bi bonds are stronger than their counterparts Lu-Pd-Bi.

Discussion

As the strength of band inversion which characterizes band topology has already been found to be connected with <Z>1,2, the scaling of the magnitude of NMR isotropic shift tensors with relativity provides new perspective into the bulk band topology of these half-Heulser compounds. The 209Bi isotropic shift tensor seems to follow a simple rule29,30,31: the theoretically predicted nontrivial phases display nearly zero or negative 209Bi shifts, while trivial phases show large positive shifts (see Table 1). For example, the measured 209Bi shifts at room temperature are −6362 ppm for LuPtBi, −4368 ppm for LuPdBi and −1748 ppm for YPtBi whereas 1200 ppm for YPdBi, 2608 ppm for ScNiBi and 3188 ppm for ScPdBi. To be noted, 209Bi NMR shifts extrapolated to zero temperatures exhibited only minor changes in magnitude compared to those taken at room temperature (except for LuPtBi for unkown reasons, see Fig. S3). In addition, there is no change in sign when extrapolated to zero temperatures for all the compounds. From the scaling of the magnitude of 209Bi shifts with relativity, it can now be understood that the change of sign in NMR shifts is due to the competitions among spin-orbit shifts (KSO) induced by relativistic SOC, scalar-relativistic shifts (KScalar) and non-relativistic shifts (KNR) from respective s or p/d electrons within the vicinity of the Fermi level. Total NMR shifts can be described as: Ktotal = KNR + KScalar + KSO21. Since the hyperfine field of s character electrons (MHz) is usually orders of magnitude of larger than that of p/d electrons (kHz), even the trace of s states at the Fermi level will contribute significantly to the total hyperfine magnetic shift via the direct Fermi-contact mechanism, which is positive just like in metals. When the s-like Г6 band is inverted and far away from the Fermi level, however, the positive Fermi-contact contribution will be lost. Furthermore, as mentioned earlier, in contrast to p3/2 states, the probability density of relativistic p1/2 electrons, does not vanish under strong spin-orbit modification in heavy atoms25. A net spin density in the ground state will then be induced by an external field because of SOC. Assuming the SOC-induced net spin density from p1/2 electrons has a spin component opposite to the external magnetic field for a topologically nontrivial half-Heusler compound, an internal magnetic field opposite to the external field at the Bi nucleus will be produced. 209Bi NMR shifts will become increasingly negative, as the occupancy of p1/2 states becomes larger, as shown in computations (see Figs 3d and 4).

Figure 4
figure 4

Electronic bulk band structures under fully relativistic calculations for (a) ScNiBi, (b) ScPdBi, (c) LuPdBi and (d) LuPtBi.

The size of the red dots indicates the degree of occupancy of p1/2-like (Γ7) bands from Bi contributions. The evolution of orbital occupancy and energy position with SOC strength will affect the magnitude of 209Bi nuclear shieldings via a Fermi-contact-like mechanism.

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Based on our NMR data we draw the conclusion that the occupancy of p1/2 states varies depending on their position controlled by SOC, which will affect the measured NMR shifts. Details of the calculations are described in the Methods section. In Fig. 4, we show the electronic bulk band structures under fully relativistic calculations for (a) ScNiBi, (b) ScPdBi, (c) LuPdBi and (d) LuPtBi, respectively, where the degree of occupancy of p1/2-like (Γ7) bands from Bi contributions are denoted as different sizes of red dots. The calculated band structures are in good accordance with former results4. During the period from (a) ScNiBi to (d) LuPtBi, we can infer that firstly, the SOC effect in the system is enhancing and pushes p1/2 states lower, away from the Fermi level and that secondly, the occupancy of relativistic p1/2 states from Bi contributions increases with SOC strength. Such differences in orbital occupancy and energy position originating from the SOC effect will affect the magnitude of 209Bi nuclear shieldings via a Fermi-contact-like mechanism26. Therefore, SOC will affect the NMR shift by tuning the relative occupancy and energy position of p1/2 states, which is at the heart of relativistic p1/2 shielding mechanism.

In conclusion, we have found the correlation of 209Bi shift tensors with relativistic SOC strength in terms of ESOC and <Z> in a class of half-Heusler R-M-Bi compounds by using NMR spectroscopy. The topologically nontrivial nature in the half-Heusler compounds is manifested as large differences in the magnitude of 209Bi NMR isotropic shifts between trivial and nontrivial half-Heusler members. This finding provides new insight into the mechanism of relativistic effects on band inversion and its role on transport properties in the topologically nontrivial half-Heusler compounds. Additional technical improvements in the NMR measurements may be directed towards understanding how the relativistic SOC effects on electronic structure manifest themselves in NMR shifts.

Methods

Sample growth

Single crystals of half-Heusler R-M-Bi (R = Sc, Y, Lu, M = Ni, Pd, Pt) were grown by using the self-flux method. The high-purity starting materials Sc, Y, Lu (ingot, 99.99%), Ni, Pd, Pt (ingot, 99.99%) and Bi (ingot, 99.99%) were mixed together in a molar ratio of 1:1:10 and afterward the mixture of pure elements were placed in an alumina crucible. This procedure was performed in a glove box filled with Ar gas, where the oxygen and humidity content were less than 0.5 ppm. The mixture was sealed inside a tantalum tube. This procedure was operated in a home-made arc furnace. The sealed tantalum tube was then sealed into an evacuated quartz tube. Crystal growth was performed in a furnace by heating the tube from room temperature up to 1150 °C over a period of 15 h and maintained for 24 h before slowly cooling down to 650 °C at a rate of 2 °C/h. The excess Bi flux was removed at 650 °C by spinning the tube in a centrifuge at rate of ~3000 r/min. After the centrifugation process, most of the excess flux was removed from crystal surfaces and the remaining flux was etched in a diluted hydrochloric acid (See Supplementary materials).

Structural characterization

The composition of the single crystal samples was determined by energy-dispersive X-ray (EDX) spectroscopy and the structure were checked by X-ray diffraction (XRD) with Cu-Kα radiation. The single-crystal orientation was checked by a standard Laue diffraction technique (See Supplementary materials).

NMR measurements and data analysis

209Bi NMR spectra of powdered single crystals were obtained by integrating the spin-echo intensity with a Bruker Avance III 400 HD spectrometer in a magnetic field of ~9.39 Tesla. Hahn echo pulse sequence was used for recording echo intensity. The first pulse length (~5 μs) is about 1/2 of the 90 degree time (tπ/2) of the samples to prevent distortion of Fourier transformed spectra. 209Bi isotropic shifts were referenced to Bi(NO3)2 aqueous nitrate solution (Bi(NO3)35H2O in 1 M nitrate solution, 8.4 ppm) (See Supplementary materials).

Band structure calculations

The band structure calculations were performed using the full-potential linear-augmented plane wave code implemented in the WIEN2K package33,34. The converged ground state was obtained using 10 000 k points in the first Brillouin zone. A combination of a modified Becke-Johnson exchange potential and the correlation potential of the local-density approximation were used to obtain the band structures37, as it predicts band gaps and the band order with favorable accuracy4. Spin-orbit coupling (SOC) was treated as a second variational procedure with scalar-relativistic orbitals as a basis, where states up to 10 Ry above the Fermi level were included in the basis expansion. Chemical shielding and EFG tensors were calculated using density functional theory (DFT), as implemented in NMR- WIEN2K34,35, where both cases for scalar and fully relativistic effective core potentials were considered.

Additional Information

How to cite this article: Zhang, X. et al. NMR Evidence for the Topologically Nontrivial Nature in a Family of Half-Heusler Compounds. Sci. Rep. 6, 23172; doi: 10.1038/srep23172 (2016).