# Heat capacity evidence for conventional superconductivity in the Type-II Dirac semi-metal PdTe

###### Abstract

We use electrical transport, magnetoresistance, and heat capacity measurements on high quality single crystals of the recently discovered superconducting Type-II Dirac semi-metal PdTe, to probe the nature of it’s superconducting phase. The magnitude of the electronic heat capacity anomaly at , the low temperature exponential dependence of the heat capacity, and a conventional phase diagram establish that the superconductivity in PdTe is conventional in nature despite the presence of a topologically non-trivial Fermi surface band which contributes to the electrical conduction.

Topological superconductors have been the focus of intense recent research Sato (). This is in part due to the possibility that these materials may host Majorana Fermion excitations Alicea2012 (); Elliot2015 () which, in addition to being of fundamental interest, can also be used in fault tolerant Quantum computation. To stabilize topological superconductivity various routes have been pursued. For example, doping Hor2010 (); Kriener2011 (); Liu2015 (); Asaba2017 (); Hor2011 (); Amit2016 (); Erickson2009 (); Balakrishnan2013 () or pressurizing Zhang2011 () a parent topological material, studying chiral spin-triplet superconductors Luke1998 (), making heterostructures of a semi-conductor with a conventional superconductor Sau2010 (); Alicea2010 (), or a topological material with a conventional superconductor Pribiag2015 (); Beenakker2016 (). Another exciting new avenue has opened up in which superconductivity has been shown to emerge in nano-scale point contacts between. Topological materials and normal metals Aggarwal2016 (); Wang2016 (); Das2016 ().

In all these routes, superconductivity is induced by some tuning like doping, pressure, proximity, or confinement. It would be ideal to look for a system in which Topological band structure and superconductivity emerge naturally and to then demonstrate the Topological character of the superconductivity.

Recently, a new family of transition metal dichalcogenide materials Pt, Pd, Te, Se) have been shown to be Type-II Dirac materials where the electronic band structure consists of a tilted Dirac cone H-Huang2016 (); Noh2017 (); Fei2017 (); Yan2017 (). This follows the discovery of Type-II Weyl materials Soluyanov2015 (); Weng2015 (); Xu2015 (); Deng2016 (); L-Huang2016 (); Jiang2017 (). Both the Type-II Weyl and Dirac Fermions observed in the above materials break Lorentz invariance and are therefore fundamentally different quasiparticles compared to the normal Type-I Dirac and Weyl Fermions discovered earlier. The study of the properties of these Type-II topological materials are therefore of immense fundamental interest and could lead to important technological applications. How conventional or fairly well understood states of matter like magnetism or superconductivity emerge in materials with Topological band-structures has been an emerging frontier area of research. In this context, PdTe is specially important since it is known to also host a superconducting state below the critical temperature K Jellinek1963 (). Topological superconductivity in PdTe is thus an exciting possibility which needs to be carefully examined.

In this work we report electrical transport, magneto-transport, and heat capacity measurements on high quality single crystals of the superconducting Type-II Dirac semi-metal PdTe to explore the possible unconventional (Topological) nature of the superconducting state. We confirm superconductivity with a critical temperature K using electrical transport measurements. From recent Shubnikov de Haas (SdH) oscillations in magneto-transport measurements we have shown that bands contribute to the transport, including a band with a non-trivial Berry phase Das2017 (). This raises the enticing possibility of Topological superconductivity in PdTe. Our heat capacity measurements demonstrate bulk superconductivity below K. The size of the superconducting anomaly in the heat capacity at is estimated to be , which is close to the value expected for a conventional weak-coupling single-gap BCS superconductor. The at low temperature shows an exponential dependence suggesting a fully gapped superconducting state. Additionally, the data in various applied magnetic fields is used to construct an – phase diagram which also shows a conventional behaviour. Thus, our measurements strongly indicate that the superconductivity in PdTe is conventional in nature despite the presence of Topologically non-trivial electrons contributing to the transport.

Experimental: Single crystals of PdTe were synthesized using a modified Bridgeman method. The starting elements, Pd powder ( purity) and Te shots (), were weighed in the atomic ratio and sealed in an evacuated quartz tube. The extra Te was taken to compensate for Te loss due to its high vapor pressure. For crystal growth, the tube with the starting materials was heated to C in h, kept there for h, and then it was slowly cooled to C over days. They were then annealed at C for days before cooling naturally. The shiny crystals of millimeter size thus obtained could be cleaved easily from the as grown boule. A typical crystal is shown on a milimeter grid in the inset of Fig. 1. The Chemical composition of crystals was verified by energy dispersive spectroscopy (EDS) on a JEOL SEM. The ratio given by EDS between Pd and Te was , showing the stoichiometric ratio of the compound. Few crystals were crushed into powder for X-ray diffraction measurements. The powder X-ray diffraction pattern confirm the phase purity of PdTe, well crystallized in the CdI-type structure with the P3m1(164) space group..

The electrical transport and heat capacity down to K were measured using the He3 option of a quantum design physical property measurement system (QD-PPMS).

Electrical Transport: Figure 1 shows the electrical resistivity versus temperature measured in zero magnetic field with a current mA applied within the crystallographic -plane. The shows metallic behaviour with K) cm and K) cm, giving a residual resistivity ratio . This is larger than reported earlier indicating that the PdTe crystals are of high quality. The lower inset in Fig. 1 shows the data below K and the abrupt drop to zero resistance below K confirms the superconductivity in PdTe.

We have recently reported Das2017 () observation of quantum oscillations in the magneto-transport measurements on PdTe crystals below K, again suggesting the high quality of the samples. For magnetic field applied perpendicular the -axis, we observed a single frequency at T in the fast Fourier transform of the dHvA data. The Berry phase for this band was estimated to be non-trivial suggesting its Topological nature. Additionally, other frequencies were observed for -axis. Thus there are multiple electronic bands including a Topological band contributing to the transport and it is unclear from just transport measurements whether the observed superconductivity itself has any unconventional Topological character.

Heat Capacity: We have therefore used heat capacity measurements to address the nature of superconductivity in PdTe. Figure 2 (a) show the heat capacity versus temperature data for PdTe between and K measured in and Oe magnetic field. A sharp anomaly at K in the data indicates that the superconductivity previously reported using only transport measurements, is bulk in nature. No anomaly is observed down to the lowest temperatures measured in Oe, suggesting that the superconductivity has been completely suppressed. This is confirmed by our heat capacity data in various magnetic field which will be presented later. The Oe data was treated as the normal state data and was fit to the expression . The fit (not shown) gave the values mJ/mol K and mJ/mol K. The lattice part was then subtracted from the data at to obtain the electronic part of the heat capacity . The electronic heat capacity divided by temperature versus is shown in Fig. 2 (b). An extremely sharp transition at the onset of superconductivity is observed at K. The normal state Sommerfeld coefficient mJ/mol K is indicated by an extrapolation (dashed line in Fig. 2 (b)) to of the normal state data. An equal entropy construction (not shown) gave almost the same K, indicating no broadening or smearing out of the superconducting transition due to sample inhomogeneties or imperfections.

The data at the lowest temperatures were fit by the expression , where is the residual Sommerfeld coefficient from the non-superconducting fraction of the sample and the second term is a phenomenological exponential decay expected for a gapped (-wave superconductor) system. The fit shown in Fig. 2 (b) as the solid curve through the data below K, gave the value mJ/mol K. With the total mJ/mol K, this suggests that of the sample volume is non-superconducting. An excellent fit of the low temperature data to an exponential dependence suggests a conventional -wave superconducting order parameter.

The magnitude of the anomaly in heat capacity at the superconducting transition is another measure of the nature (weak or strong coupling, single or multi-gap) of superconductivity. From the data in Fig. 2 (b) we estimate , which is close to the value expected for a conventional, weak-coupling, single gap BCS superconductor. This further supports the conventional nature of superconductivity in PdTe.

Heat capacity measurements at various magnetic fields are shown in Fig. 3. As expected, the superconducting transition temperature is monotonically suppressed to lower temperatures and its magnitude becomes smaller at higher fields. From an equal entropy construction for the data at each , we extract the at that and use it to draw a critical-field versus temperature () phase diagram. The phase diagram is shown in Fig. 4 and follows a conventional behaviour expected for a BCS superconductor. In particular, we were able to fit the data with the phenomenological expression , with the critical field , and the critical temperature as fit parameters. The fit shown as the solid curve through the data in Fig. 4 gave the values Oe and K, respectively.

Summary and Discussion: PdTe is an interesting material where a superconducting state below K coexists with a Topological band structure. Specifically, PdTe has previously been shown to be a Type-II Dirac semi-metal raising the possibility of hosting a Topological superconducting state. In this study we have used thermodynamic measurements on high quality single crystals of PdTe to probe the nature of the superconductivity. Our heat capacity measurements confirm bulk superconductivity at K and show that the anomaly at is characterized by the ratio which is close to the value expected for a weak-coupling, single-band BCS superconductor. The electronic contribution to the heat capacity at the lowest temperatures shows an exponential dependence which points to a gapped -wave superconductivity. Additionally, the critical field versus temperature phase diagram shows a behaviour expected for a conventional superconductor. Therefore, all our results strongly indicate that inspite of the presence of a Topological band in the electronic band-structure of PdTe which contributes to the transport properties, the superconductivity in PdTe is completely conventional and has no Topological character.

Acknowledgments.– We thank the X-ray and SEM facilities at IISER Mohali.

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