Fully gapped superconducting state in Au₂Pb: A natural candidate for topological superconductor

In recent years, the topological materials have attracted much attention because of their novel quantum states [1–3]. Among those topological materials, the topological superconductors (TSCs) are of particular interest. The TSCs have a full gap in the bulk and Majorana fermion states on the surface [2]. The TSCs are very important since they possess potential application to topological quantum computation due to the non-Abelian statistics of Majorana fermions [4]. Experimentally, there are several routes to obtain a TSC candidate. The first one is to artificially fabricate topological insulator/superconductor heterostructures [5,6]. The second one is to dope a topological material. For example, the Cu-intercalated Bi₂Se₃ shows superconductivity below 3.8 K [7], which was considered as a TSC candidate [8–11]. The third one is to pressurize a topological material. Pressure-induced superconductivity was observed in some topological insulators, such as Bi₂Se₃ [12], Bi₂Te₃ [13], and Sb₂Te₃ [14]. Superconductivity was also found in the three-dimensional Dirac semimetal Cd₃As₂ under a tip or hydropressure [15–17]. Note that the pressure-induced superconductivity usually coincides with a change in crystal structure and the material may be no longer topological [12,17]. To further confirm that a TSC candidate is indeed a TSC, it is essential to identify the Majorana fermion states on the surface. Unfortunately, no consensus has been reached on this important issue yet [18].

Besides these routes, in some rare cases, a natural TSC candidate may exist if a stoichiometric superconductor manifests topological properties under ambient pressure. One recent example is the noncentrosymmetric superconductor PbTaSe₂ with $T_c = 3.72\ \text{K}$ [19]. Angle-resolved photoemission spectroscopy (ARPES) experiments and first-principle calculations revealed topological nodal-line semimetal states and associated surface states in PbTaSe₂ [20,21]. The thermal conductivity, specific heat, and London penetration depth measurements all demonstrated a fully gapped superconducting state in it [22–24], which is also required for a TSC.

More recently, it was argued that the cubic Laves phase Au₂Pb $(T_c \approx 1.2\ \text{K})$ may be another natural TSC candidate [25,26]. Electronic band structure calculations predicted that cubic Au₂Pb has a bulk Dirac cone at room temperature [25]. With decreasing temperature, Au₂Pb undergoes structural phase transitions, and only the orthorhombic phase remains below 40 K [25,27]. Their calculations showed that the structure transition gaps out the Dirac spectrum in the high-temperature phase, and results in a low-temperature nontrivial massive 3D Dirac phase with $Z_2 = -1$ topology [25]. In ref. [26], the first-principles calculations also point to the nontrivial topology of the orbital texture near the dominant Fermi surfaces, which suggests the possibility of topological superconductivity. To check whether Au₂Pb is indeed a TSC, it will be very important to determine its superconducting gap structure first.

In this letter, we investigate the superconducting gap characteristics of the Au₂Pb single crystal, by means of the ultra-low-temperature specific heat and thermal conductivity measurements. Our analysis shows that the electronic specific heat can be described by the two-band s-wave model. Furthermore, a negligible $\kappa_0/T$ in zero field and a slow field dependence of $\kappa_0/T$ at low field are revealed by the thermal conductivity measurements. These results suggest that Au₂Pb has a fully gapped s-wave superconducting state in the bulk.

Single crystals of Au₂Pb were grown by a self-flux method [26]. The as-grown single crystals have a plate-like shape, and the largest natural surface was determined as (111) plane at room temperature [26]. The sample for transport measurements was cut to a rectangular shape of dimensions $2.5 \times 1.0\ \text{mm}^2$ in the (111) plane, with a thickness of 0.10 mm perpendicular to this natural surface. Contacts were made directly on the sample surfaces with silver paint, which were used for both resistivity and thermal conductivity measurements. The typical contact resistance is less than $100\ \text{m}{\Omega}$ at low temperature. The resistivity was measured in a ⁴He cryostat from 300 K to 2 K, and in a ³He cryostat down to 0.3 K. The thermal conductivity was measured in a dilution refrigerator, using a standard four-wire steady-state method with two RuO₂ chip thermometers, calibrated in situ against a reference RuO₂ thermometer. Magnetic fields were perpendicular to the natural surface and the heat current. Note that the natural surface is no longer the (111) plane for the low-temperature orthorhombic, distorted phase of Au₂Pb [25]. To ensure a homogeneous field distribution in the sample, all fields were applied at a temperature above T_c. The low-temperature specific heat was measured from 0.1 to 3.5 K in a physical property measurement system (PPMS, Quantum Design) equipped with a small dilution refrigerator.

Figure 1(a) shows the resistivity of the Au₂Pb single crystal in zero field. The three discontinuities at 97, 50, and 40 K are related to the structural phase transitions, as reported previously [25–27]. From the inset of fig. 1(a), the width of the resistive superconducting transition (10–90%) is 0.16 K, and the T_c defined by $\rho = 0$ is 1.05 K. The $\rho(T)$ curve between 1.5 and 2.5 K is quite flat, which extrapolates to a residual resistivity $\rho_0 = 5.71\ \mu\Omega\,\text{cm}$. Thus, the residual resistivity ratio $(\text{RRR} = \rho(295\ \text{K})/\rho_0)$ is about 7.5.

Zoom In Zoom Out Reset image size

To determine the upper critical field H_c2(0) of Au₂Pb, we measured the low-temperature resistivity of the sample in various magnetic fields up to 100 mT, as shown in fig. 1(b). The temperature dependence of $H_{c2}(T)$, defined by ρ = 0 on the resistivity curves in fig. 1(b), is plotted in fig. 1(c). The dashed line is a guide to the eye, from which we estimate H_c2(0) ≈ 19.1 mT. A slightly different H_c2(0) does not affect our discussion on the field dependence of $\kappa_0/T$ below. Note that if the T_c is defined at the onset of the superconducting transition, the estimated H_c2(0) will be 86.2 mT, which is comparable to that of ref. [25].

Figure 2(a) shows the low-temperature specific heat of the Au₂Pb single crystal down to 0.1 K in zero field. A significant jump due to the superconducting transition is observed at $T_c \approx 1.16\ \text{K}$. At low temperature, the normal-state specific heat C_p can be described by $C_p = C_{en} + C_{lat}$, in which $C_{en} = \gamma_nT$ is the electronic contribution and $C_{lat} = \beta T^3 + \delta T^5$ is the lattice contribution. The solid line in fig. 2(a) is the best fit to the data between 1.2 and 1.7 K, yielding $\gamma_n = 2.8\ \text{mJ mol}^{-1}\text{K}^{-2}$, $\beta = 1.1\ \text{mJ mol}^{-1} \text{K}^{-4}$, and $\delta = 0.24\ \text{mJ mol}^{-1} \text{K}^{-6}$. With the normal-state electronic specific heat deduced from this fit, the entropy in the normal and superconducting states agrees to 1% at T_c, as shown in fig. 2(b). From the relation $\theta_D = (12\pi^4RZ/5\beta)^{1/3}$, where R is the molar gas constant and Z is the total number of atoms in one unit cell, the Debye temperature $\theta_D = 174\ \text{K}$ is estimated. Note that the Sommerfeld coefficient $\gamma_n$ and the Debye temperature $\theta_D$ obtained from our data are slightly larger than those reported previously [25,27]. From the zero-field specific-heat data, the thermodynamic critical field was determined to be 8 mT by calculating the free-energy difference between the normal and superconducting states.

Zoom In Zoom Out Reset image size

Figure 2(c) displays the temperature dependence of the electron specific heat $C_e/T = C_p/T - C_{lat}/T$. At the superconducting transition, the specific-heat jump $\Delta C_e/\gamma_nT_c$ is estimated to be about 1.40, which is close to the weak-coupling BCS prediction of 1.43. In ref. [25], the specific heat of Au₂Pb was measured down to 0.4 K. Here, we measured it down to lower temperature $(\sim 0.1\ \text{K})$, which enables us to quantitatively analyze its behavior. We note that $C_e/T$ almost saturates and extrapolates to a negligible value at ultra-low temperature, yielding a residual value $\gamma_r = 0.084\ \text{mJ mol}^{-1} \text{K}^{-2}$. Taken at face value, a superconducting volume fraction $(\gamma_n-\gamma_r)/\gamma_n \approx 97\%$ is estimated. First, we simply fit $C_e/T$ in the superconducting state with the BCS α-model, as shown in fig. 2(c). Since electronic structure calculations of Au₂Pb showed that there are several bands which cross the Fermi level along the $\Gamma\text{-}K$ line [25], there are likely more than one superconducting gaps. On the basis of a phenomenological two-gap model within the conventional BCS framework [28,29], the T-dependence of $C_e/T$ including the whole T region below T_c can also be fitted by using two BCS gaps. In this fit, the specific heat is calculated as the sum of the contributions from two bands by assuming independent BCS temperature dependence of the two s-wave superconducting gaps. The magnitudes of the two gaps at the T = 0 limit are introduced as adjustable parameters, α₁ = Δ₁(0)/k_BTc and α₂ = Δ₂(0)/k_BTc, together with the quantity $\gamma_i/\gamma_n$ ($i = 1$, 2), which is the weight of the total electron density of states (EDOS) for each band. As shown in fig. 2(c), the two-band fit with $\alpha_1 = 1.41$ and $\alpha_2 = 5.25$ reproduces the specific heat very well. The superconducting gap ratio Δ₁(0)/Δ₂(0) ≈ 0.27 is obtained. The weights are 8% and 92% for the two bands with the small and large gaps, respectively. This specific-heat result indicates a fully gapped superconducting state in Au₂Pb.

The ultra-low-temperature thermal conductivity measurement is another bulk technique to probe the superconducting gap structure [30]. In fig. 3, we present the temperature dependence of the thermal conductivity for the Au₂Pb single crystal in zero and magnetic fields. The thermal conductivity at very low temperature usually can be fitted to $\kappa/T = a + bT^{\alpha-1}$ [31,32], where the two terms aT and $bT^{\alpha}$ represent contributions from electrons and phonons, respectively. In order to obtain the residual linear term $\kappa_0/T$ contributed by electrons, we extrapolate $\kappa/T$ to $T = 0\ \text{K}$. Because of the specular reflections of phonons at the sample surfaces, the power α in the second term is typically between 2 and 3 [31,32].

Zoom In Zoom Out Reset image size

We first examine the accuracy of the Wiedemman-Franz law in the normal state of Au₂Pb. In fig. 3(b), the fit of the data in $H = 17\ \text{mT}$ gives $\kappa_{0}/T = 4.33\ \text{mW K}^{-2} \text{cm}^{-1}$. This value meets the Wiedemann-Franz law expectation $L_0/\rho_0(17\ \text{mT}) = 4.29\ \text{mW K}^{-2}\,\text{cm}^{-1}$ nicely, with the Lorenz number $L_0 = 2.45 \times 10^{-8}\ \text{W}\,\Omega\,\text{K}^{-2}$ and $\rho_0(17\ \text{mT}) = 5.71\ \mu\Omega\,\text{cm}$. Since the $H = 20\ \text{mT}$ data are overlapped with the $H = 17\ \text{mT}$ data within the experimental uncertainty, we take $H = 17\ \text{mT}$ as the bulk H_c2(0) of Au₂Pb, which is slightly lower than that estimated from resistivity measurements. The verification of the Wiedemann-Franz law in the normal state shows the reliability of our thermal conductivity measurements.

In zero field, the fitting gives the residual linear term $\kappa_0/T = - 2.6\ \pm\ 6\ \mu \text{W K}^{-2} \text{cm}^{-1}$, as seen in fig. 3(a). Considering our experimental uncertainty $\pm 5\ \mu \text{W K}^{-2} \text{cm}^{-1}$, the $\kappa_0/T$ is essentially zero. For nodeless superconductors, there are no fermionic quasiparticles to conduct heat as $T \rightarrow 0$, since all electrons become Cooper pairs. Therefore, there is no residual linear term $\kappa_0/T$, as seen in conventional s-wave superconductors Nb and InBi [33,34]. However, for unconventional superconductors with nodes in the superconducting gap, the nodal quasiparticles will contribute a substantial $\kappa_0/T$ in zero field. For example, $\kappa_0/T = 1.41\ \text{mW K}^{-2} \text{cm}^{-1}$ for the overdoped d-wave cuprate superconductor Tl₂Ba₂CuO_6+δ (Tl-2201), which is about 36% of its $\kappa_{N0}/T$ [35], and $\kappa_0/T = 17\ \text{mW K}^{-2}\text{cm}^{-1}$ for the p-wave superconductor Sr₂RuO₄, which is about 9% of its $\kappa_{N0}/T$ [36]. Therefore, a negligible $\kappa_0/T$ of Au₂Pb in zero field also suggests a fully gapped superconducting state.

The field dependence of $\kappa_0/T$ can give more information about the superconducting gap structure [30]. Between $H = 0$ and 17 mT, we fit all the curves and obtain the $\kappa_0/T$ for each magnetic field, as shown in fig. 3(b). The normalized $\kappa_0/T$ of Au₂Pb as a function of $H/H_{c2}$ is plotted in fig. 4. For comparison, the data of the clean s-wave superconductor Nb [33], the dirty s-wave superconducting alloy InBi [34], the multiband s-wave superconductor NbSe₂ [37], and an overdoped d-wave cuprate superconductor Tl-2201 [35], are also plotted. For a clean s-wave superconductor with a single gap, $\kappa_0(H)/T$ should grow exponentially with the field, as observed in Nb [33]. While for the s-wave InBi in the dirty limit, the curve is exponential at low H, crossing over to a roughly linear behavior closer to H_c2 [34]. In the case of NbSe₂, the distinct $\kappa_0(H)/T$ behavior was well explained by multiple superconducting gaps with different magnitudes [37].

Zoom In Zoom Out Reset image size

In fig. 4, the field dependence of $\kappa_0(H)/T$ for Au₂Pb grows slightly faster than that of the dirty s-wave superconductor InBi. Note that the internal field is actually larger than the applied field when considering the demagnetization factor [38]. This effect is more significant at low fields, thus the $\kappa_0(H)/T$ of Au₂Pb may be close to that of InBi. Since we do not know enough parameters, such as the Fermi velocity, to estimate the mean free path of the carriers in Au₂Pb, we cannot judge whether it is in the clean or dirty limit. In case that it is in the clean limit, such a $\kappa_0(H)/T$ behavior may result from multiple superconducting gaps. As we mentioned earlier, electronic structure calculations of Au₂Pb showed that there are several bands [25]. Analysis of electronic specific heat also reveals two superconducting gaps with the amplitudes Δ₁(0)/k_BT_c = 1.41 and $\Delta_2(0)/k_BT_c = 5.25$, respectively. Note that the weight of the small gap is low (8%), which may be the reason why the $\kappa_0(H)/T$ of Au₂Pb grows much slower than that of NbSe₂.

So far, PbTaSe₂ and Au₂Pb are the only two natural TSC candidates. Our results demonstrate that Au₂Pb has a similar fully gapped superconducting state to PbTaSe₂. A full superconducting gap is a necessary condition for TSCs, but is also the criterion for a conventional s-wave superconductor. To judge which the case is, it is essential to know their topological properties. While the topological band structure and surface states have been verified in PbTaSe₂, detailed ARPES experiments on Au₂Pb are highly desired. Note that recent scanning tunneling microscopy (STM) experiments on PbTaSe₂ observed a full superconducting gap in zero field, and the zero-energy bound states at superconducting vortex cores in magnetic field [39]. However, the thermal broadening at $T = 0.26\ \text{K}$ makes it impossible to discriminate the ordinary Caroli-de Gennes Matricon (CdGM) bound states and Majorana bound states [39]. Therefore, more STM experiments, such as spin-polarized STM [6], are needed to determine whether these two natural TSC candidates are TSCs or conventional s-wave superconductors.

In summary, the superconducting gap structure of Au₂Pb has been studied by the ultra-low-temperature specific heat and thermal conductivity experiments. The analysis of the electronic specific heat suggests two s-wave superconducting gaps with the amplitudes $\Delta_1(0)/k_BT_c = 1.41$ and $\Delta_2(0)/k_BT_c = 5.25$. Furthermore, a negligible $\kappa_0/T$ in zero field and the field dependence of $\kappa_0/T$ also suggest nodeless superconducting gaps. These results indicate that the second natural TSC candidate Au₂Pb has a fully gapped superconducting state in the bulk.

We thank Lei Shu, Jian Zhang, Zhaofeng Ding, and Cheng Tan for helpful discussions. This work is supported by the Natural Science Foundation of China, the Ministry of Science and Technology of China (National Basic Research Program No. 2012CB821402 and No. 2015CB921401), Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, and STCSM of China (No. 15XD1500200). The work in Peking University was supported by National Basic Research Program of China (Grant Nos. 2013CB934600 and 2012CB921300), the Open Project Program of the Pulsed High Magnetic Field Facility (Grant No. PHMFF2015002), Huazhong University of Science and Technology, and Open Research Fund Program of the State Key Laboratory of Low-Dimensional Quantum Physics, Tsinghua University under Grant No. KF201501.