Single full gap with mixed type-I and type-II superconductivity on surface of the type-II Dirac semimetal PdTe2 by point-contact spectroscopy

TL;DR

This study uses point-contact spectroscopy to investigate superconductivity in PdTe2, revealing coexistence of type-I and type-II behaviors likely due to surface inhomogeneities related to topological states.

Contribution

First detailed PCS analysis showing mixed type-I and type-II superconductivity on PdTe2 surface, linked to topological surface states.

Findings

Superconducting gap follows BCS temperature dependence.

Observation of both first-order and gradual transitions in magnetic field.

Evidence of surface inhomogeneity affecting superconducting type.

Abstract

We report our point-contact spectroscopy (PCS) study on the superconducting state of the type-II Dirac semimetal PdTe$_2$ with a superconducting transition temperature $T\rm_c \sim$ 1.65 K. Both mechanical- and soft- PCS differential conductance curves at 0.3 K show a consistent double-peak structure and they can be perfectly fitted by a single s-wave gap based on the Blonder-Tinkham-Klapwijk model. The gap follows a typical Bardeen-Cooper-Schrieffer temperature behavior, yielding $\Delta_0 \sim$ 0.29 meV and 2$\Delta_0$/$k\rm_{B}$$T\rm_{c}$ = 4.15 in the strong coupling regime. A sudden suppression of the superconducting gap in magnetic field around $H{\rm_{c1}}$ $\sim$ 130 Oe is observed for most point-contacts on PdTe$_2$, characteristic of a first-order transition for type-I superconductor in field. However, for other contacts, a smooth evolution of the PCS conductance persists up to $H{\rm_{c2}}$ $\sim$ 600 Oe, signaling a local type-II superconductivity. The observed admixture of type-I and type-II superconductivity can possibly arise from an inhomogeneous electron mean free path on the surface of PdTe$_2$ due to its topological surface states.

Full text

Soon after the discovery of topological insulators (TIs), topological superconductors (TSCs) have also attracted intensive attention in the community because of the nontrivial topology for Bogoliubov quasiparticles in TSCs and electronic bands in TIs Qi and Zhang (2011); Hasan and Kane (2010); Ando and Fu (2015); Sato and Ando (2017); Read and Green (2000); Qi et al. (2009); Schnyder et al. (2008); Kitaev (2009). Moreover, Majorana fermions are expected to be hosted on the surface or edge of TSCs, complying with non-Abelian statistics and playing a decisive role in quantum computation Nayak et al. (2008); Kitaev (2003); Beenakker (2013). One strategy to realize TSCs is to induce superconductivity (SC) in topological materials, such as topological insulator, Weyl and Dirac semimetals, by taking advantage of the nontrivial topology of the electronic bands Sato (2009); Fu and Berg (2010); Qi et al. (2010); Yamakage et al. (2012); Lu et al. (2015); Kobayashi and Sato (2015); Bednik et al. (2015). For example, topological SC has been claimed for the topological surface states with proximity-induced SC at the interface of a TI/SC heterostructure or on the surface of some iron-based superconductors, where Majorana zero modes are argued to exist at their vortices or edges Fu and Kane (2008); Xu et al. (2015, 2014a); Wang et al. (2012); Xu et al. (2014b, b); Zareapour et al. (2012); Wang et al. (2013); Zhang et al. (2018); Wang et al. (2018); Liu et al. (2018).

Recently, the transition-metal dichalcogenide compound PdTe2 has been confirmed to be a type-II Dirac semimetal with a tilted Dirac cone below the Fermi energy by the angle resolved photoemission spectroscopy (ARPES) and there exists a spin-polarized topological surface state Noh et al. (2017); Clark et al. (2018). De Haas-van Alphen (dHvA) experiments also evidence the nontrivial Berry phase for one of the Fermi-surface pockets, probably a hole pocket from the tilted Dirac cone Fei et al. (2017); Zheng et al. (2018). On the other hand, superconductivity in PdTe2 with a transition temperature $T\rm_{c}$ $\sim$ 1.7 K has been known for a long time, serving as a promising candidate for stoichiometric TSCs Leng et al. (2017). However, heat capacity, London penetration depth, tunneling junction and STM measurements all support a conventional fully-gapped s-wave superconductor for PdTe2 Amit and Singh (2018); Teknowijoyo et al. (2018); Salis et al. (2018); Sirohi et al. (2019); Clark et al. (2018); Das et al. (2018). Moreover, recent electrical transport and heat capacity measurements reveal a puzzling discrepancy of its critical field for the superconducting state with 3000 Oe and 250 Oe, respectively. It is difficult to be explained by either a general Saint-James-de Gennes (SJdG) surface critical field or filamentary superconductivity Leng et al. (2017); Amit and Singh (2018). One possible scenario proposed by Siroshi et al. is that bulk PdTe2 is a typical type-I superconductor and there exists a mixing of type-I and type-II superconductivity on its surface Leng et al. (2017); Amit and Singh (2018); Sirohi et al. (2019). More experiments are thus desired to elucidate its exact superconducting nature for the type-II Dirac semimetal PdTe2.

In this article, we have applied both mechanical and soft point-contact spectroscopy (PCS) to investigate the superconducting gap of single crystalline PdTe2. Our PCS conductance curves at 0.3 K can be well fitted by a single gap s-wave Blonder-Tinkham-Klapwijk (BTK) model Blonder et al. (1982) and the temperature evolution of the extracted superconducting gap follows a conventional Bardeen-Cooper-Schrieffer (BCS) behavior, yielding a gap $\Delta_{0}\sim$ 0.29 meV with 2$\Delta_{0}$/k\rm_{B}$$T\rm_{c} = 4.15. Even though some point-contacts show an abrupt suppression of the SC gap by magnetic field at 130 Oe, characteristic of a type-I superconductor, some other contacts exhibit a smooth evolution in field with a typical type-II superconducting behavior below a critical field of 600 Oe, suggesting an inhomogeneous mixing of type-I and type-II SC on the PdTe2 crystal surface.

PdTe2 single crystals were grown by the flux method: High purity Pd and Te with the molar ratio 1:4 were put into an aluminum crucible and then sealed in a quartz tube. The sample was heated up to 800 ∘C and then cooled to 500 ∘C slowly. The excess Te were removed by centrifuge at this temperature and single crystals PdTe2 were left at the bottom of the crucible. Before experiment, the samples were cleaved to expose fresh surface at ambient condition. Mechanical PCS (MPCS) in a needle-anvil configuration was employed, where an electrochemically-etched sharp gold tip was gently engaged on the crystal surface by piezo-controlled nano-positioners. In comparison, soft PCS (SPCS) contacts on PdTe2 were prepared on the sample surface by attaching a 30 $\mu$m diam gold wire with a silver-paint drop at the end yielding a total contact diameter around $\sim$ 50-100 $\mu$m. Hundreds of parallel nanoscale junctions are assumed between individual silver particles and the crystal surface for SPCS Daghero and Gonnelli (2010); Sasaki et al. (2011). In the case of MPCS, even though parallel nanoscale channels should also exist, the total contact diameter from the sharp tip end is less than 5 $\mu$m, much smaller than that of SPCS. The PCS differential conductance curves as a function of bias voltage, $G$($V$), were recorded with the conventional lock-in technique in a quasi-four-probe configuration. Oxford cryostat with a He3 insert was used to cool the whole PCS setup down to 0.3 K and magnetic field was applied along the crystal c-axis up to 1000 Oe to suppress SC.

Figure 1 shows eight representative PCS differential conductance curves $G$($V$) on PdTe2 at the lowest temperature 0.3 K with the left and right panel for MPCS and SPCS, respectively. All the conductance curves show a common double-peak structure around $\pm$ 0.3 meV and no dip feature at high bias is present in any curve, ensuring the ballistic nature of our contacts Sheet et al. (2004). A single gap s-wave BTK model can perfectly fit all experimental curves as shown by the black lines in Fig. 1 and the extracted superconducting gap $\Delta$ values at 0.3 K are scattered in the range of 0.276 and 0.301 meV, yielding 2$\Delta$/k\rm_{B}$$T\rm_{c} = 4.00 - 4.36 in a strong-coupling regime. We notice that our PCS gap values are consistent with those reported by STM ($\sim$ 0.289 meV) Sirohi et al. (2019); Das et al. (2018), but larger than the weak-coupling value estimated from bulk specific heat and penetration depth studies Amit and Singh (2018); Teknowijoyo et al. (2018). The discrepancy of gap size is suspected to be associated with the spin-polarized topological surface state, because both PCS and STM are more sensitive to the SC on surface rather than the bulk SC. However, its exact origin needs further clarification Lv et al. (2017); Mizushima et al. (2014); Guan et al. (2019). For optimal fittings, the smearing parameter $\Gamma$ is much smaller than the superconducting gap and comparable between MPCS and SPCS, while the barrier strength parameter Z ranges from $\sim$ 0.5 to 0.78 for different contacts in Fig. 1 possibly from the Fermi velocity mismatch between the tip and sample Blonder et al. (1982); Blonder and Tinkham (1983).

Figure 2(a) and (b) show the temperature evolution of conductance curves $G$($V$) from 0.3 to 1.6 K for MPCS and SPCS on PdTe2, respectively. With increased temperatures, the double peaks are smeared to a single peak and finally disappear at the superconducting $T\rm_{c}\sim$ 1.65 K for PdTe2. We note that the $T\rm_{c}$ determined by the zero-bias conductance (ZBC) curves for both mechanical and soft contacts are around 1.65 K, consistent with the resistive $T\rm_{c}$ as shown in Fig. 2(c). The extracted SC gap values from an optimal BTK fitting are shown in Fig. 2(d) and it follows the conventional BCS temperature behavior. As in the inset of Fig. 2(d), the fitting parameters Z and $\Gamma$ for both MPCS and SPCS show little change with temperature, indicating an ideal ballistic contact Huang et al. (2008).

For the MPCS field dependence, the conductance curves $G$($V$) show an interesting behavior as in Fig. 3(a): They barely change but overlap with each other below 75 Oe, and the conductance peaks are abruptly suppressed in intensity at 100 Oe and quickly disappear around 125 Oe, while the peak positions stay almost unshifted in field. The MPCS zero-bias conductance as a function of field displays a sudden drop at 130 Oe as in Fig. 3(c), strongly supporting a type-I superconductor for PdTe2 as reported by magnetization, heat capacity and STM measurements Leng et al. (2017); Amit and Singh (2018); Sirohi et al. (2019). In contrast, the SPCS conductance curves for PdTe2 as in Fig. 3(b) show a gradually reduced peak intensity in field, whereas the peak positions have no change before the suppression of SC at 130 Oe. We note that, for type-I superconductors, an intermediate state will emerge far below its critical field when the field is perpendicular to the sample plane with a large demagnetization factor. In such a case, normal state domains can coexist with the SC domains in space and progressively replace the superconducting volume Prozorov (2007); Kozhevnikov et al. (2014); Prozorov et al. (2005). In order to mimic this process, a modified two-component BTK model has been applied to fit the conductance curves with both contributions from the normal and SC regions, $G(V)$ = \omega$$G_{Normal} + (1-$\omega$)G\rm_{BTK}$$(V), where $G_{Normal}$ is just flat for the normal state, G\rm_{BTK}$$(V) is the standard BTK curve for the SC regions, and the fitting parameter $\omega$ denotes the spectra weight from the normal state. The SPCS conductance curves at different fields can be well fitted by this modified BTK model as shown in Fig. 3(b) and the parameters Z and $\Gamma$ change little with field as in MPCS. The normal state spectra weight $\omega$ shows a roughly linear increase with field as in Fig. 3(d) and vividly illustrates a monotonic increase of the normal state volume in the intermediate state. Nonetheless, the SC gap for the SC regions keeps constant in field up to 100 Oe for PdTe2, indicative of a type-I superconductor. This modified BTK model can also be applicable to the MPCS case as in Fig. 3(d): A sudden increase of the $\omega$ value from zero implies normal state domains entering the contact area, only when the field gets close to $H{\rm{}_{c1}}$. The systematic difference between MPCS and SPCS is due to its much smaller total contact area in MPCS so that the tip only probes a single superconducting domain for MPCS in low fields.

For MPCS, we have collected a set of ZBC curves as a function of field at 0.3 K, $G_{0}(H)$, for contacts M1-M16 as shown in Fig. 4(a). Each contact is obtained once after the tip is withdrawn, moved to another position by xy positioners and gently engaged on the sample with the z positioner, where M1-M16 are from the same tip but not numerically ordered as the experimental sequence. While type-I SC for PdTe2 can been established by the sudden drop at 130 Oe in ZBC for contacts M7-M16, we notice that other ZBC curves M1-M6 show a smooth evolution into the normal state at a critical field $H\rm_{c2}$ around 600 Oe. The continuous transition for M1-M6 with a much larger critical field than the bulk $H{\rm{}_{c1}}$ $\sim$ 130 Oe suggests a local type-II superconductivity at these contacts, consistent with the STM and resistivity measurements for current along c axis Sirohi et al. (2019); Fei et al. (2017). A corresponding conductance curve $G(V)$ at 0.3 K has an obvious dip structure at higher bias voltages as in Fig. 4(c), implying the contact not in a pure ballistic limit and the mean free path $l$ is believed smaller than the diameter of each contact channel Sheet et al. (2004). It is intriguing that no additional dips are observed for any contact among M7- M16 with type-I SC, while dips are commonly observed for contacts M1-M6 with type-II SC. Some SPCS conductance curves also exhibit a smooth evolution of the SC gap with field as in Fig. 4(b) and the inset of Fig. 4(d). Its conductance at a fixed bias voltage 0.2 meV shows a similar continuous type-II behavior at 600 Oe after a first-order-like drop at 130 Oe as in the inset of Fig. 4(d), signaling a spacial coexistence of type-I and type-II SC in the soft contact area (We note that the zero-bias conductance for our SPCS contacts available is comparable to the value in the normal state, and so the conductance at a fixed bias voltage 0.2 meV rather rather than zero bias is chosen to easily identify the critical field). A weak dip structure at high bias is also observed with the conductance curve deviating from its BTK fitting as in Fig. 4(d). We notice that a similar contact-dependence for PCS spectra has been reported in a recent study on the non-centrosymmetric superconductor Re6Zr and it is ascribed to the surface superconductivity as well Parab et al. (2019).

In general, the Ginzburg-Landau (GL) parameter $\kappa$ = $\frac{\lambda}{\xi}$ is defined to describe type-I ($\kappa$ $<$ $\frac{1}{\sqrt{2}}$ ) and type-II ($\kappa$ $>$ $\frac{1}{\sqrt{2}}$) superconductors, where $\lambda$ is the penetration depth and $\xi$ the coherence length of superconductors Tinkham (2004). It was claimed that the $\kappa$ in PdTe2 is around 0.34 $<$ $\frac{1}{\sqrt{2}}$, categorizing PdTe2 as a type-I superconductor Leng et al. (2017). In order to explain the emergence of type-II SC behavior, we follow the arguments by Siroshi et al. and consider the Pippard nonlocal electrodynamics $\frac{1}{\xi}=\frac{1}{\xi_{0}}+\frac{1}{l}$ and $\lambda=\lambda_{L}\sqrt{1+\frac{\xi_{0}}{l}}$, where $\xi_{0}$ and $\lambda_{L}$ is the BCS coherence length and London penetration depth with $l$ $\rightarrow\infty$, respectively. Following the relations $\xi_{0}=\frac{0.18\hbar^{2}k_{F}}{k_{B}T_{c}m^{\ast}}$ and $\lambda_{L}=\sqrt{\frac{m^{\ast}}{\mu_{0}ne^{2}}}$, if we assume the carrier density n = 5.5 $\times 10^{27}$ m*-1Leng et al. (2017), the effective mass $m^{\ast}$ $\approx 0.3m_{e}$Dunsworth (1975); Fei et al. (2017) ($m_{e}$ is the free electron mass), and the Fermi wave number $k_{F}=\sqrt[3]{3\pi^{2}n}$ = 5.5$\times 10^{9}$ m-1*, we can get $\lambda_{L}=39$ nm and $\xi_{0}=1800$ nm for PdTe2. Since $\kappa=\lambda_{L}(\frac{1}{\xi_{0}}+\frac{1}{l})\sqrt{1+\frac{\xi_{0}}{l}}$, $\kappa$ as a function of $l$ is plotted in Fig. 5 as the pink line. We note that the mean free path $l$ for bulk PdTe2 is estimated $l\approx$ 341 nm from the Drude model $l=\frac{3\pi^{2}\hbar}{\rho e^{2}k{{}_{F}^{2}}}$, which makes $\kappa\sim$ 0.341 in the type-I superconductor regime. A reduced mean free path $l$ on some surface regions can cause an enhanced $\kappa$ and thus tune it into a type-II SC with $l\leq$ 200 nm. We thus argue that the dip structure at high bias frequently observed for contacts with the type-II SC behavior implies a greatly reduced mean free path $l$, which makes local surface SC regions enter into the type-II regime.

As for the critical field, according to the GL theory, $H{\rm{}_{c1}}=\frac{\phi_{0}}{2\sqrt{2}\pi\mu_{0}\lambda\xi}=\frac{\phi_{0}}{2\sqrt{2}\pi\mu_{0}\lambda_{L}\sqrt{\xi_{0}}}\sqrt{\frac{1}{\xi_{0}}+\frac{1}{l}}$ and $H{\rm{}_{c2}}=\frac{\phi_{0}}{2\pi\mu_{0}\xi^{2}}=\frac{\phi_{0}}{2\pi\mu_{0}}(\frac{1}{\xi_{0}}+\frac{1}{l})^{2}$. The dependence of $H{\rm{}_{c1}}$ and $H{\rm{}_{c2}}$ on the mean free path $l$ is shown in Fig. 5 with red and blue segments, respectively. A mean free path $l$ $\sim$ 341 nm estimated from the Drude model will give an $H{\rm{}_{c1}}$ around 83 Oe not so off from our PCS value 130 Oe for type-I SC behavior. As shown in Fig. 5, a slight variation of the mean free path $l$ has no dramatic effect on $H{\rm{}_{c1}}$ in the type-I regions, consistent with the fixed critical fields $\sim$ 130 Oe from different PCS contacts.

For the type-II superconductor behavior among our PCS contacts, a critical field around 600 Oe is consistently observed without a large variation either. However, Fig. 5 indicates a slight change of the mean free path $l$ on surface should cause a wide distribution of $H{\rm{}_{c2}}$ for $l\leq$ 200 nm. Our observations probably argues against the mixture of type-I and type-II SC simply due to the disorder effect on surface. We notice that a variation of local density of states is frequently observed on the surface of several topological materials such as Bi2Se3, Bi2Te3 and BiSbTeSe2, where charge puddles are proposed from a local fluctuation of the chemical potential when the Dirac cone is close to it Knispel et al. (2017); Borgwardt et al. (2016); Zhang et al. (2009); Beidenkopf et al. (2011). The accumulated charge puddles on the surface should increase the local charge carriers substantially for the low-carrier-density semimetal and thus reduce the mean free path $l$ with $l=\frac{3\pi^{2}\hbar}{\rho e^{2}k{{}_{F}^{2}}}$ and $k_{F}=\sqrt[3]{3\pi^{2}n}$. Once the carrier density $n$ goes beyond 4000 $\times 10^{25}$ m*-3*, the reduced mean free path $l$ $\sim$ 100 nm inside different puddles doesn’t change too much as shown in the inset of Fig. 5, giving a critical field $H{\rm{}_{c2}}$ between 400 -700 Oe. This crude model can roughly explain the disparate $H{\rm{}_{c}}$ values between type-I and type-II SC in our PCS results.

Our MPCS and SPCS spectra show a reproducible double peak structure with the absence of a zero-bias conductance peak and a perfect fitting with a single gap s-wave BTK model strongly supports a conventional s-wave paring gap in the type-II Dirac semimetal PdTe2 as evidenced by other measurements Amit and Singh (2018); Teknowijoyo et al. (2018); Salis et al. (2018); Sirohi et al. (2019); Clark et al. (2018); Das et al. (2018). On the other hand, SC with a full gap is claimed for topological surface states to host Majorana zero modes in vortices for some iron-based superconductors Zhang et al. (2018); Wang et al. (2018); Liu et al. (2018). As a type-I superconductor, it is thus difficult to check Majorana zero modes for PdTe2 without vortex lattice. However, a mixture of type-I and type-II SC on the surface offers a chance to search for Majorana fermions in PdTe2, if any. For PdTe2, this mixing is attributed to an inhomogeneous electron mean free path $l$ on the surface. More studies are needed to illuminate the relationship between this puzzling inhomogeneous behavior and topological surface states in PdTe2 especially by STM measurements.

In conclusion, we have observed a single full superconducting gap $\Delta$ from both MPCS and SPCS on the type II Dirac semimetal PdTe2 with $\Delta_{0}\sim$ 0.29 meV and 2$\Delta_{0}$/k\rm_{B}$$T\rm_{c} = 4.15 in the strong coupling regime. However, the field-dependent conductance curves suggest a mixture of type-I and type-II SC with distinct critical fields probably due to an inhomogeneous electron mean free path on the surface of PdTe2. More careful studies are called for to address its origin and possible relations with topological surface states in PdTe2.

We are grateful for valuable discussions with L. Li, Y. Zheng and Y. Zhou. Our work was supported by the National Key Research & Development Program of China (Grant No. 2016FYA0300402, 2017YFA0302901 and 2017YFA0303101) and the National Natural Science Foundation of China (Grant No. 11674279, 11774399 and 11374257). X.L. would like to acknowledge support from the Zhejiang Provincial Natural Science Foundation of China (LR18A04001) and Y.G.S. acknowledges support from Beijing Natural Science Foundation (Z180008).