Band Inversion in Topologically Nontrivial Half-Heusler Bismuthides

Jan 12, 2015 - postulated inversion of the electronic bands near the Fermi level. For the former two ... a gapless surface state that is protected by ...
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Band Inversion in Topologically Nontrivial Half-Heusler Bismuthides: 209 Bi NMR Study B. Nowak, O. Pavlosiuk, and D. Kaczorowski* Institute of Low Temperature and Structure Research, Polish Academy of Sciences, P.O. Box 1410, 50-950 Wrocław, Poland ABSTRACT: 209Bi nuclear magnetic resonance (NMR) was measured in half-Heusler bismuthides ScPdBi, LuPdBi, and LuPtBi and discussed in conjunction with the NMR data recently reported for YPdBi and YPtBi. The observed 209Bi NMR spectra exhibit low intensity because of short nuclear transverse relaxation time (T2) governed by sizable structural disorder. Remarkably, the NMR frequency shifts are positive for topologically trivial ScPdBi and YPdBi, yet strongly negative for topologically nontrivial YPtBi, LuPtBi, and LuPdBi with postulated inversion of the electronic bands near the Fermi level. For the former two compounds, the spin−lattice relaxation rate (1/T1) is very large and seems correlated with the strength of spin−orbit coupling. The study demonstrates the capability of NMR spectroscopy to probe the band inversion in putative topological insulators.

1. INTRODUCTION Rare-earth (R) based ternary compounds RTX (T = Ni, Pd, Pt; X = Sb, Bi) belong to a large family of the so-called half-Heusler phases, which crystallize with a noncentrosymmetric cubic structure of the MgAgAs-type.1 A few of them are presently a subject of exceptional interest due to indications from ab initio electronic structure calculations of possible band inversion near the Fermi level and formation of a topologically nontrivial state, characterized by insulating band gap in their bulk, while having a gapless surface state that is protected by time-reversal symmetry.2−10 The topology of the electronic bands in halfHeusler compounds is defined by so-called band inversion strength (BIS), Δ, defined as an energy difference between Γ6 and Γ8 energy levels at the Γ symmetry point in the Brillouin zone, i.e., Δ = Γ6 − Γ8 (note that in ref 5 this is defined as Δ = Γ8 − Γ6). The compounds with positive values of Δ are topologically trivial, while those with negative Δ values are topologically nontrivial. It is generally believed that the band inversion in these systems is a consequence of strong spin− orbit coupling (SOC).11−14 However, some authors argued that in many materials scalar relativistic effects on the s electrons of heavy elements may play an essential role in the band inversion, instead of SOC.15 The electronic structure calculations, based on a fullpotential linearized-augmented-plane-wave (FPLAPW) method with modified Becke−Johnson local-density-approximation (MBJLDA) of exchange-correlation potential, indicated that half-Heusler bismuthides YPtBi, LaPtBi, LuPtBi, and LuPdBi exhibit the band inversion effect.5,6 In contrast, the same calculations revealed ScPdBi and YPdBi to be topologically trivial.5,6 Most interestingly, all the band-inverted RTBi materials were experimentally found to be superconductors at low temperature.16−21 The superconductivity emerges there from semiconducting or semimetallic normal states, governed by a strong SOC effect, and hence, it may possibly bear a nontrivial topological nature.20,21 © XXXX American Chemical Society

In view of the theoretical results and the observed superconducting behaviors, it is highly tempting to verify the actual topological character of the RTBi compounds. However, angle-resolved photoemission spectroscopy (ARPES) experiments performed on single-crystalline RPtBi (R = Gd, Dy, Lu) did not provide any clear evidence for the presence of topological insulating (TI) states.22 In turn, the interpretation of TI-like features recently observed in the electrical transport of YPdBi and LuPdBi,21,23 such as large linear magnetoresistance or weak antilocalization (WAL) effect, is seriously hampered by the difficulty in reliably unraveling the contributions due to the surface states from those due to semiconducting bulk. Here, we postulate that a very appropriate experimental tool for examining the band inversion in the half-Heusler compounds, a prerequisite condition for their putative TI behaviors, is nuclear magnetic resonance (NMR). As a localprobe method, NMR is sensitive to the electronic band structure in the vicinity of the Fermi level. In the past, 125Te NMR was performed on Hg1−xCdxTe that is presently considered as an archetypal two-dimensional (2D) TI system.24 In turn, the bulk properties of well-recognized 3D-TI systems, Bi2Se3, Bi2Te3, and some ordered ternaries, were studied by 77 Se and 125Te NMR,25−27 as well as by 209Bi NMR.28,29 Recently, we communicated the results of our 209Bi NMR measurements of YPdBi and YPtBi, which revealed considerable differences between these two compounds,30 theoretically predicted as being topologically trivial and nontrivial, respectively.5,6 In topologically trivial systems the conduction band has predominantly s-character, while in nontrivial materials the conduction bands have p- or d-symmetry. Thus, the observed positive 209Bi NMR shifts imply Bi-s Fermi Received: November 18, 2014 Revised: January 9, 2015

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MgAgAs-type of their crystal structures, characteristic of halfHeusler phases.1 In the noncentrosymmetric crystallographic unit cell (space group F4̅3m, No. 216), the particular atoms occupy the positions 4a (0,0,0), 4b (1/2,1/2,1/2), 4c (1/4,1/ 4,1/4), all having the same site-symmetry 4̅3m. As an example of the obtained PXRD results, the diffraction pattern of LuPtBi is shown in Figure 1. The cubic lattice parameter obtained from

contact interaction, while negative ones hint at dominance of chemical shielding contribution, rather than Bi-p core polarization. Our key findings have been confirmed by an independent group.31 In the present work, we performed 209Bi NMR studies on two other half-Heusler systems considered nontrivial, namely, LuPtBi and LuPdBi, as well as on a topologically trivial material ScPdBi (ScPtBi does not form).5,6 The NMR results obtained for the entire RTBi (R = Sc, Y, Lu) series, discussed here in conjunction with the 121,123Sb NMR data found for the corresponding antimonides,32,33 allow us to demonstrate convincingly the unique capability of NMR technique to probe the band inversion effect in these compounds.

2. EXPERIMENTAL METHODS Single crystals of LuPdBi and LuPtBi were grown from Bi flux. The elemental constituents (purities: Lu, 99.9 wt %; Pd and Pt, 99.9 wt %; Bi, 99.999 wt %) were taken in atomic ratio 1:1:13, placed in alumina crucibles, and sealed in quartz tubes under argon atmosphere. The ampules were heated up to 1200 °C at a rate of 50 °C/h and kept at this temperature for 5 h. Subsequently, they were slowly cooled down to 500 °C with a rate of 3 °C/h. At this step, the excess of bismuth was separated from the products using a commercial centrifuge. The soobtained single crystals had dimensions up to about 0.5 × 0.5 × 0.5 mm3. Polycrystalline sample of ScPdBi was prepared by arc-melting the appropriate amounts of the constituents (purities: Sc, 99.9 wt %; Pd and Bi, see above) on a water-cooled copper crucible in ultrapure argon atmosphere. The buttons were flipped over and remelted several times to ensure good homogeneity. The final weight losses were below 1%. Then, the ingot was wrapped with tantalum foil, sealed in an evacuated (vacuum 10−4 Torr) quartz tube, and annealed at 800 °C for 2 weeks. The obtained samples were characterized at room temperature by powder X-ray diffraction (PXRD) carried out using an X’pert Pro PANanalitical diffractometer with Cu Kα radiation. The crystal structure refinements and the theoretical PXRD pattern calculations were done employing the FULLPROF program package.34 The chemical compositions of the samples were examined on an FEI scanning electron microscope equipped with an EDAX Genesis XM4 spectrometer. The materials were found homogeneous, essentially free of foreign phases, with the chemical compositions very close to the ideal ones. NMR measurements were performed on fine-pulverized specimens between 50 and 293 K using a Bruker Avance DSX 300 spectrometer operating at a field of 7.05 T, equipped with a temperature controller ITC-503 (Oxford Instruments Co Ltd.). The 209Bi NMR spectra were obtained by Fourier transforming the free induction decay (FID) signal following single radio frequency (RF) pulse. Quadrature detection and extended phase cycling procedures were used. The line widths were defined as a full width at half-maximum (fwhm). According to the IUPAC unified δ scale,35 the 209Bi NMR frequency shifts were determined with reference to the respective value of 209Ξ defined as the ratio of the isotope-specific frequency to that of 1 H in tetramethylsilane (TMS) in the same magnetic field.

Figure 1. Powder X-ray diffraction pattern of single-crystalline LuPtBi. The observed and calculated data are marked by dots and solid line, respectively. Difference pattern is shown in the bottom of the figure. Vertical bars denote the positions of Bragg reflections.

this data is a = 6.578(1) Å, in very good agreement with the literature values: 6.574(1) Å determined for polycrystalline sample36 and 6.578 Å found for single-crystalline sample.20 Similarly, the refined lattice parameter for our ScPdBi specimen, a = 6.432(1) Å, is very close to that reported in the literature (6.435(1) Å; ref 36). As regards LuPdBi, the PXRD data yielded the lattice parameter a = 6.565(1) Å, in perfect accord with the literature value 6.566(1) Å determined for polycrystalline sample,36 however, being markedly different from the lattice parameter of 6.63 Å, stated by Xu et al. for their single crystals.21 In the MgAgAs-type unit cell of the half-Heusler bismuthides, the Bi atoms are expected to occupy the sites with tetrahedral symmetry, and thus, no quadrupolar effects should appear in the 209Bi NMR spectra, despite the large quadrupolar moment of the nuclei I(209Bi) = 9/2. However, as discussed in ref 30, an inherent property of the RTBi phases is structural imperfection, particularly arising from site interchange. This feature brings about generation of electric field gradient (EFG) at the nominally tetrahedral Bi atom site that considerably influences the intensity of the NMR signals due to short spin−spin relaxation time T2. For all three bismuthides studied in the present work, ScPdBi, LuPtBi, and LuPdBi, a single resonance signal was found. By analogy with the results reported before for YPtBi and YPdBi,30,31 and also for YbPtBi,37 the observed NMR line can be attributed to the 209Bi nuclei. As an example, the NMR spectra obtained for LuPtBi at T = 293 K and T = 100 K are presented in Figure 2a (the spectrum recorded at 100 K has lower signal-to-noise ratio since considerably fewer scans were collected). The line widths were found to be practically independent of temperature indicating that the measured sample is well-ordered and the amount of possible structural defects is fairly small. The line widths fwhm ≈20−25 kHz are

3. RESULTS AND DISCUSSION The X-ray diffraction characterization of the obtained samples of ScPdBi, LuPdBi, and LuPtBi unambiguously confirmed the B

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procedure for YPtBi and YPdBi: T1 = 26 ms and T1 = 66 ms, respectively.30 Since the NMR line in LuPtBi is narrow, the fast relaxation cannot be due to high carrier concentration in the measured sample. Instead, as discussed in ref 30, it reflects the band inversion due to strong SOC, in line with the large and negative frequency shift established for this material. In semiconductors and semimetals the principal contributions to the frequency shift of NMR signal are the chemical shift (δCS) and the Knight shift (δKS), describing the interactions between bond electrons and the observed nucleus, and the free electrons and the observed nucleus, respectively. To distinguish between these two type of interactions a study of the shifts should be made as a function of temperature and concentration of carriers (n- or p-type), followed by extrapolation of the shift to zero carrier concentration. This procedure is time-consuming, and the boundary between the ntype and p-type (zero Knight shift) is not yet established for Bibased half-Heusler compounds. Moreover, the electron populations of the conduction bands and the free carrier concentration are expected to be more strongly dependent on temperature than the diamagnetic and paramagnetic contributions to the chemical shift. For these reasons, the term Knight shift is frequently used loosely to include all sources of the shift in nuclear resonance.28,29 The so-defined Knight shift (K ≡ δ) is composed of a temperature independent orbital part Korb = K0 ≡ δCS, and a temperature dependent spin part Kspin = K(T). The total shift measured with respect to appropriate standard recommended by IUPAC35 is given by K = K0 + K(T). Figure 2c shows the temperature dependence of the 209Bi NMR frequency shift in LuPtBi. The δ values shift to higher frequencies with rising temperature, reflecting an increase in the spin susceptibility, which can be attributed to thermally driven increase in the number of carriers (nH), also responsible for enhancement in the relaxation rate 209(1/T1). The experimental data can be very well-described by the function δ = K = K0 + bTλ, where b is a constant and ∂K/∂T > 0. Large and negative K0 = −7930 ppm obtained by extrapolation to T = 0 may be attributed to strong chemical shielding characteristic of topologically nontrivial material that exhibits band inversion effect (see below). It is worthwhile mentioning that similar features (negative line shifts of 125Te nuclei and ∂K/∂T > 0) have been observed in an archetypal two-dimensional (2D) TI system Hg1−xCdxTe.24 On the other hand, in a standard topologically trivial half-Heusler semiconductor such as TiPtSn, the 195Pt Knight shifts were found to weakly shift to lower frequencies with increasing temperature (∂K/∂T < 0).38 From ∂K/∂T > 0 one may conclude that the thermally excited charge carriers in LuPtBi predominantly have s character, with a positive s-hyperfine constant. Evidently, a classical temperature dependence of the NMR shift, K(T) ∝ nH/kBT, does not apply for LuPtBi. Furthermore, the observed functional variation of δ with λ = 2, i.e., notably stronger than that expected for zero-gap systems, where nH ∝ T3/2 (ref 39), is not clear at present and calls for further studies. It must be noted that the above analysis of the 209Bi NMR line shift is only qualitative since the carrier concentration in the measured sample of LuPtBi was not determined. Also, the actual influence of the charge density on the NMR parameters is not known. Nevertheless, for the Y-based bismuthides it was argued30 that the carrier concentration and especially the distribution of native carriers must be weak since the spread in the values of fwhm, δ, and T1 obtained for samples from different batches was reasonably small. Moreover, nearly the

Figure 2. 209Bi NMR data of topologically nontrivial LuPtBi. (a) Line shape spectrum at T = 293 K and T = 100 K. (b) Recovery of the nuclear magnetization for 209Bi nuclei at T = 293 K. The solid line stands for the exponential fit. (c) Temperature dependence of the frequency shift. The solid curve represents the power law fit discussed in the text.

much larger than the calculated dipolar one. Similar line broadening was found for YPdBi and ScPdBi. Along with the arguments given in ref 30, this effect can be attributed to quadrupolar interactions operative in the presence of chemical disorder in nominally cubic crystal structure (vacancies, antisite positions, etc.). Most remarkably, in the two Lu-based compounds the 209Bi NMR frequency shift is negative. It is as large as δ = −6050 ppm for LuPtBi and δ = −4350 ppm for LuPdBi that can be compared with δ = −1780 ppm found for YPtBi.30 In contrast, the frequency shift in ScPdBi is positive and equal to δ = 3150 ppm that is larger than δ = 1130 ppm derived for YPdBi.30 As demonstrated in Figure 2b, the spin−lattice relaxation data of LuPtBi can be fit with a three-parameter magnetization recovery function M(t) = M(∞)[1 − C exp(−t/T1)], where M(t) and M(∞) represent, respectively, the signal intensity at time t after saturation and at thermal equilibrium, and T1 is the spin−lattice relaxation time. The linear dependence of the recovery function over two decades indicates very weak dispersion of the relaxation rate. Treating M(∞) and C ≈ 1 as adjustable parameters, T1 of 4.7 ms was obtained. This value is distinctly smaller from those determined via similar C

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4. CONCLUSIONS Negative 209Bi NMR frequency shifts observed for the RTBi compounds, accompanied by very short spin−lattice relaxation times, appear to be good indicators of the band inversion Γ6/Γ8 in these materials that gives rise to their topologically nontrivial character. In contrast, topologically trivial bismuthides RTBi without a band inversion (alike topologically trivial RTSb antimonides) always exhibit positive NMR frequency shifts.

same NMR data were reported for YPtBi and YPdBi by other authors.31 All these facts support the conclusion that the contribution from carriers, while definitely present, can be regarded as a minor factor governing the frequency shifts in the RTBi phases. The key message from our study is visualized in Figure 3. The 209 Bi NMR frequency shifts of the half-Heusler

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AUTHOR INFORMATION

Corresponding Author

*Phone: +48 71 395 42 58. Fax: +48 71 344 10 29. E-mail: D. [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was financially supported by the National Science Centre (Poland) under research grant 2011/01/B/ST3/04466.

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REFERENCES

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Figure 3. Stick NMR spectra at T = 293 K. (a) 121Sb NMR frequency shifts for the half-Heusler antimonides ScPdSb and ScPtSb (labeled Sc),32 YPdSb and YPtSb (labeled Y),33 and LuPdSb (ref 40) and LuPtSb (ref 33) (labeled Lu), consecutively ordered from the left to the right. (b) 209Bi NMR shifts of the respective half-Heusler bismuthides. For the sake of clarity, the scale (in ppm) for Sb resonance was expanded. (c) Schematic visualization of the principal difference in the electronic band structure in topologically trivial and topologically nontrivial systems.

bismuthides form two separate subgroups with opposite signs, and this division correlates perfectly well with the predicted topological character of a given compound. Namely, all the RTBi phases theoretically described as topological semimetals with inverted electronic bands (note Figure 3c), i.e., YPtBi, LuPdBi, and LuPtBi,5,6 exhibit negative 209Bi NMR shifts, whereas those identified as topologically trivial, i.e., ScPdBi and YPdBi,5,6 show positive NMR frequency shifts. Furthermore, in the nontrivial regime, an increasing magnitude of negative shift is accompanied by distinct shortening of the 209Bi spin−lattice relaxation time. Shown in Figure 3 are also the 121Sb NMR frequency shifts derived for the half-Heusler antimonides,32,33,40 each being topologically trivial according to the band structure calculations.5,6 Remarkably, the numerical values of 121Sb NMR shifts are all positive and rather weakly dispersed from +490 to +1495 ppm at ambient temperature when going from Lu- to Sc-based compounds. It is also worth noting that the spin− lattice relaxation time of the 121Sb nuclei in all these antimonides is equal to about 100 ms, nearly independent of the rare-earth component. D

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(33) Nowak, B.; Kaczorowski, D. Nonmetallic Behaviour in HalfHeusler Phases YPdSb, YPtSb and LuPtSb. Intermetallics 2013, 40, 28−35. (34) Rodriguez-Carvajal, J. Recent Advances in Magnetic Structure Determination by Neutron Powder Diffraction. Physica B 1993, 192, 55−69. (35) Harris, R. K.; Becker, E. D.; Cabral de Menezes, S. M.; Goodfellow, R.; Granger, P. NMR Nomenclature: Nuclear Spin Properties and Conventions for Chemical Shifts. Solid State Nucl. Magn. Reson. 2002, 22, 458−483. (36) Haase, M. G.; Schmidt, T.; Richter, C. G.; Block, H.; Jeitschko, W. Equiatomic Rare Earth (Ln) Transition Metal Antimonides LnTSb (T=Rh, Ir) and Bismuthides LnTBi (T=Rh, Ni, Pd, Pt). J. Solid State Chem. 2002, 168, 18−27. (37) Reyes, A. P.; Le, L. P.; Heffner, R. H.; Ahrens, E. T.; Fisk, Z.; Canfield, P. C. 209Bi NMR in the Heavy-Electron System YbBiPt. Physica B 1995, 206&207, 332−335. (38) Grykałowska, A.; Nowak, B. Nuclear Spin-Lattice Relaxation in Narrow Gap Semiconductors TiPtSn and ZrPtSn. Intermetallics 2007, 15, 1479−1482. (39) Tsidilkovski, I.M. Electron Spectrum of Gapless Semiconductors; Springer: New York, 1996; Vol. 116. (40) The NMR data for LuPdSb (fwhm = 6.5 kHz, δ = +673 ppm, T1 = 92 ms at T = 293 K) were obtained in the course of the present study for a polycrystalline sample.

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