Shubnikov-de Haas oscillations, weak antilocalization effect and large linear magnetoresistance in the putative topological superconductor LuPdBi

We present electronic transport and magnetic properties of single crystals of semimetallic half-Heusler phase LuPdBi, having theoretically predicted band inversion requisite for nontrivial topological properties. The compound exhibits superconductivity below a critical temperature $T_{\rm c}=1.8\,$K, with a zero-temperature upper critical field $B_{\rm c2}\approx2.3\,$T. Although superconducting state is clearly reflected in the electrical resistivity and magnetic susceptibility data, no corresponding anomaly can be seen in the specific heat. Temperature dependence of the electrical resistivity suggests existence of two parallel conduction channels: metallic and semiconducting, with the latter making negligible contribution at low temperatures. The magnetoresistance is huge and clearly shows a weak antilocalization effect in small magnetic fields. Above about 1.5 T, the magnetoresistance becomes linear and does not saturate in fields up to 9 T. The linear magnetoresistance is observed up to room temperature. Below 10 K, it is accompanied by Shubnikov-de Haas oscillations. Their analysis reveals charge carriers with effective mass of $0.06\,m_e$ and a Berry phase very close to $\pi$, expected for Dirac-fermion surface states, thus corroborating topological nature of the material.

. XRD pattern for powdered single crystals of LuPdBi.
It is worth noting that despite the aforementioned divergence in the values of the lattice parameter a, the X-ray diffractogram of LuPdBi presented in Ref. 3 appears very similar to our result displayed in Fig. S1. Small differences in the relative intensities of a few Bragg peaks [e.g., (311), (222), (420)] might be attributed to a certain level of atomic disorder in the sample studied by Xu at al., since our PXRD data were refined to a small value of the residual Bragg factor R = 4.61%, with full occupancies of all the crystallographic positions and no structural disorder (see also the discussion presented in Ref. 4).
The crystal structure of the studied samples of LuPdBi was also checked by the single crystal X-ray diffraction method. Details on the performed refinement are gathered in Table S1. The positional data and the equivalent isotropic displacement parameters are given in Table S2. The so-refined lattice parameter a = 6.5683(10)Åis very close to that derived from the PXRD data, and thus significantly smaller than the result reported for single-crystalline LuPdBi by Xu at al. 3  Table S1. Crystal data and structure refinement for LuPdBi. Table S2. Atomic coordinates and equivalent isotropic displacement parameters U (eq) for LuPdBi. U (eq) is defined as one third of the trace of the orthogonalized U ij tensor.
Energy-dispersive X-ray spectroscopy Examples of the EDX spectrum and the SEM image obtained for the single crystals of LuPdBi is presented in Fig.  S2. The chemical composition derived from these data is Lu 32.00(32) Pd 33.93(33) Bi 34.07(34) , in a very good accord with the ideal equiatomic one. Figure S2. SEM-EDX results for single-crystalline LuPdBi.

WAL parameters
The temperature dependencies of the weak antilocalization parameters ηe 2 2π 2h and L ϕ (together with their uncertainties d( ηe 2 2π 2h ) and dL ϕ ) obtained from fitting the Hikami-Larkin-Nagaoka (HLN) formula (Eq. 1 in the main text) to the experimental σ(B) data of LuPdBi are collected in Table S3. When both parameters were free to vary in fitting procedure, the values given in second and fourth columns of the Table S3 were obtained. However the L ϕ varied with temperature in irregular way and both parameters were strongly dependent on each other, as shown by statistical dependency values in fifth column. Thus we could not consider all these values of ηe 2 2π 2h and L ϕ as reliable. Reasoning that the WAL model is most adequate at lowest temperature, where the contribution from the bulk semiconducting channel is the smallest, we repeated the fitting with the value of ηe 2 2π 2h fixed at the value derived at T = 2.5 K. Resulting values of L ϕ are collected in sixth column of the Table. It is worth noting that their errors are now smaller by about one order of magnitude. These values decrease monotonously with increasing temperature, which seems to reflect the increasing of the bulk channel contribution.  For more clarity we plotted the values of ηe 2 2π 2h and L ϕ resulting from both fitting procedures, versus temperature, as shown in Fig. S3. The temperature variation of the phase coherence length Lϕ| η(T =2.5K) (resulting from the HLN fit with ηe 2 2π 2h fixed at the value derived at T = 2.5 K.) can be roughly described by the formula given in Ref. 3 Fig. S3 by the solid red curve), however such a fit yields the electron-phonon interaction parameter close to zero.

(represented in
The ηe 2 2π 2h parameter derived in Ref. 3 was about an order of magnitude smaller than that obtained on our sample, whereas corresponding values of L ϕ were about twice as large as our Lϕ| η(T =2.5K) data. This is rather a good agreement, taking into account different electronic character of the samples examined in our work and that described in Ref. 3. It seems to be a common feature of half-Heusler compounds that the values of parameter −η are of order 10 5 − 10 6 , thus much larger than 1/2 expected for 2D TI systems. Apart from LuPdBi it has very recently been observed in samples of LuPtSb, 5 YPtBi and LuPtBi. 6 Authors of all these papers attributed this fairly unexpected result to contributions from bulk or side wall conductivity channels.

Hall parameters
In concert with the previous report, 3 the Hall carrier concentration in the single crystal investigated in our work was found temperature independent below 50 K and then to increase with increasing temperatures. However, in the entire temperature range covered, the magnitude of n H determined in our study is approximately four times smaller than that estimated by Xu et al. (their result was n H 4.8 × 10 19 cm −3 at low temperatures). Remarkably, our findings are completely different as regards the Hall carrier mobility. For our sample, µ H was found to gradually decrease with increasing temperature from the value 2404 cm 2 V −1 s −1 at T = 2.5 K to 573 cm 2 V −1 s −1 at 300 K, while Xu et al. reported an increase from 330 to 380 cm 2 V −1 s −1 in the same temperature interval.

Heat capacity
The striking result reported in Ref. 3 is the occurrence of a pronounced λ-shaped anomaly in the specific heat of LuPdBi that coincides with the superconducting transition. Consequently, Xu et al. ascribed the superconductivity in their single crystals to the bulk. This result markedly differs from the behavior of our single crystals, where no visible feature in C(T ) was found near T c , despite the formation of superconducting state was convincingly proved by means of the magnetic and electrical transport measurements. This finding led us to the conclusion that the superconductivity in our samples of LuPdBi is confined to the surface, with the superconducting condensate volume negligibly small as compared to the bulk. The result obtained by Xu et al. differs also from the properties of closely related systems LuPtBi and YPtBi, 7,8 for which the specific heat was found featureless at the superconducting transition. For the latter compound we recently confirmed the lack of any anomaly at T c in our own investigation carried out on high-quality single crystals. 9 The observed discrepancy might be related to the different electronic character of the single crystals studied in our work and those investigated in Ref. 3. Comparison of the temperature-dependent electrical resistivity suggests that our samples are much more semimetallic than the predominantly semiconducting one reported by Xu et al. In this context, however, the observed over-an-order-of-magnitude discrepancy between the Sommerfeld coefficient γ = 0.75mJ/mol K 2 derived in the present work and γ = 11.9mJ/mol K 2 found by Xu et al. becomes most surprising. Here, again, one should note that the electronic specific heat of the related semimetallic nonmagnetic bismuthides, like LuPtBi and YPtBi (Refs. 7-10), and antimonides, like LuPtSb, YPdSb and YPtSb (Ref. 11) is similar to that established by us for LuPdBi.