Two-dimensional (2D) superconductivity has been extensively studied in recent years to explore exotic quantum phenomena, such as high-temperature superconductivity and various quantum phase transitions.^[1–3] There are several ways to realize the 2D superconductivity as described below.

(i) It can be realized in ultra-thin films grown by molecular beam epitaxy method, e.g., in monolayer FeSe/SrTiO₃ (T_c = 65 K),^[1] ultra-thin Ga films ( ),^[2] and highly crystalline Pb thin films (T_c = 1.5 K).^[4,5]

(ii) The 2D superconductivity can be found in some particular interfaces or surfaces, e.g., the LaAlO₃/SrTiO₃ interface (T_c = 0.2 K),^[6] the interface of an insulator and a metal La₂CuO₄/La_1.55Sr_0.45CuO₄ (T_c exceeds 50 K),^[7] and the surface of MoS₂ thin flake by ionic liquid gating (the maximum T_c = 10.8 K).^[8,9]

(iii) By vertically putting two graphene sheets together with certain twisted angles, both a Mott-like insulating state and a superconducting state (T_c ≈ 1.7 K) arise from the electrons localized in these 2D superlattices.^[10,11]

(iv) An relatively easier way is to mechanically exfoliate thin flakes directly from bulk van der Waals (vdW) quasi-2D superconductors. Bi₂Sr₂CaCu₂O_8+x (Bi-2212) and NbSe₂ have been successfully cleaved to atomic thin sheets with different thicknesses, in which superconductivity has been observed (T_c = 82 K for half-unit-cell Bi-2212 and T_c = 3.1 K for monolayer NbSe₂, respectively).^[12–15] In this context, it is important to find more attractive vdW quasi-2D superconductors.

Very recently, a new gold-based quasi-2D superconductor AuTe₂Se_4/3 has been discovered.^[16] Despite that metastable Au_1−xTe_x (0.6 < x < 0.85) compounds are superconductors with T_c between 1.5 K and 3.0 K,^[17,18] calaverite AuTe₂ is a non-superconducting compound.^[19] By introducing Se atoms into AuTe₂, electrons from Te have been attracted by the Se anions, and superconductivity with was found in AuTe₂Se_4/3.^[16] Furthermore, it manifests the 2D Berezinsky–Kosterlitz–Thouless (BKT) topological transition based on the measurements of standard I–V curves around T_c, which indicates the superconductivity is mainly confined in the ab plane.^[16] This is consistent with the fact that the interaction of interlayer is Van der Waals force in AuTe₂Se_4/3, and the AuTe₂Se_4/3 single crystal can be very easily exfoliated into thin flakes by Scotch tape.^[16] Therefore, AuTe₂Se_4/3 is a new vdW quasi-2D superconductor.

To understand the superconducting pairing mechanism of a new superconductor, it is crucial to know the superconducting gap structure first. In this work, we measure the ultra-low-temperature thermal conductivity of AuTe₂Se_4/3 single crystal down to 100 mK, to determine its superconducting gap structure. A negligible residual linear term κ₀/T in zero field and a slow field dependence of κ₀/T at low field are revealed, which suggest multiple nodeless superconducting gaps in this interesting vdW quasi-2D superconductor. This agrees with our previous band structure calculations in Ref. [16].

Single crystals of AuTe₂Se_4/3 were grown by the self-flux method.^[16] The obtained single crystals have ribbon shape with shining mirror-like surfaces, as shown in the inset of Fig. 1(a). The largest surface was identified as the ab plane by x-ray diffraction (XRD) measurement, as shown in Fig. 1(a), in which only (00l) Bragg peaks show up. The sample for transport measurements was cut to a rectangular shape of 2.20 mm × 0.60 mm in the ab plane, with a thickness of 0.20 mm perpendicular to the ab plane. Four gold electrodes were fabricated on the sample by thermal evaporation to ensure a good contact at low temperature. Four silver wires were then attached to the electrodes with silver paint, which were used for both resistivity and thermal conductivity measurements, with electrical and heat currents in the ab plane. The contacts are metallic with typical resistance of 5 mΩ at 0.3 K. The thermal conductivity was measured in a dilution refrigerator, using two RuO₂ chip thermometers, calibrated in situ against a reference RuO₂ thermometer. All the magnetic fields for resistivity and thermal conductivity measurements were applied perpendicular to the ab plane. To ensure a homogeneous field distribution in the sample, all fields were applied at temperature above the temperature T_c.

Fig. 1.

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Fig. 1. (color online) (a) X-ray diffraction pattern of AuTe2Se4/3 single crystal. Only (00l) Bragg peaks show up, demonstrating that the largest surface is ab plane. Inset: Optical image of our AuTe2Se4/3 single crystals. (b) The temperature dependence of resistivity at zero field for AuTe2Se4/3 single crystal. Top inset: The fit of the normal-state resistivity to the Fermi-liquid behavior ρ(T) = ρ0 + AT2 from 3 K to 15 K. Bottom inset: The resistive superconducting transition at low temperature.

Figure 1(b) shows the temperature dependence of the resistivity ρ(T) for AuTe₂Se_4/3 single crystal at zero field. From the bottom inset of Fig. 1(b), the width of the resistive superconducting transition (10%–90%) is less than 0.05 K. The T_c defined by ρ = 0 is 2.70 K, which is consistent with our previous magnetic susceptibility measurement.^[16] From 3 K to 15 K, the normal-state resistivity can be well fitted by the Fermi-liquid behavior ρ(T) = ρ₀ + AT², as shown in the top inset of Fig. 1(b). The obtained residual resistivity ρ₀ = 87.3 μΩ · cm gives the residual resistivity ratio (RRR) ρ (290 K)/ρ₀ ≈ 3.69.

In order to estimate the upper critical field H_c2(0), the low-temperature resistivity of the AuTe₂Se_4/3 single crystal was measured in magnetic fields up to 0.7 T, as shown in Fig. 2(a). The temperature dependence of H_c2(T), defined by ρ = 0, is plotted in Fig. 2(b). One can see an apparently linear temperature dependence of H_c2. With a linear fit to the data, H_c2(0) ≈ 0.49 T is roughly estimated. Actually, the linear temperature dependence of H_c2 in Fig. 2(b) is very interesting. It may be explained by a two-band Fermi surface topology as in MgB₂,^[20–22] or an unconventional superconducting state as in heavy-fermion compound UBe₁₃.^[23] Density function theory calculation shows there are four bands in AuTe₂Se_4/3,^[16] therefore its linear behavior of H_c2(T) likely originates from multiple-band Fermi surface topology.

Fig. 2.

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	Fig. 2. (color online) (a) Low-temperature resistivity of AuTe2Se4/3 single crystal in magnetic fields up to 0.7 T. (b) Temperature dependence of the upper critical field Hc2(T), defined by ρ = 0. The dashed line is a linear fit to the data, which points to Hc2(0) ≈ 0.49 T.

Ultra-low temperature thermal conductivity measurement is an established bulk technique to probe the superconducting gap structure.^[24] The temperature dependence of thermal conductivity for the AuTe₂Se_4/3 single crystal in zero and magnetic fields is plotted in Fig. 3. The measured thermal conductivity contains two contributions, κ = κ_e + κ_p, which come from electrons and phonons, respectively. In order to separate the two contributions, all the curves are fitted to κ/T = a + bT^α−1 below 0.3 K.^[25,26] The two terms aT and bT^α represent contributions from electrons and phonons, respectively. The residual linear term κ₀/T ≡ a is obtained by extrapolating κ/T to T = 0 K. Due to the specular reflections of phonons at the sample surfaces, the power α in the second term is typically between 2 and 3.^[25,26]

Fig. 3.

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Fig. 3. (color online) Low-temperature thermal conductivity of AuTe2Se4/3 single crystal (a) in zero field, and (b) in magnetic fields applied perpendicular to the ab plane. The solid lines represent the fits to κ/T = a + bTα − 1 for the data in different H. The dashed line is the normal-state Wiedemann–Franz law expectation L0/ρ0(0.4 T), with the Lorenz number L0 = 2.45 × 10−8 W · Ω · K−2 and ρ0(0.4 T) = 87.3 μΩ · cm.

In zero field, the fitting gives κ₀/T = −9.9 ± 11 μW · K⁻² · cm⁻¹ and α = 2.41, as seen in Fig. 3(a). Comparing with our experimental error bar ±5 μW · K⁻² · cm⁻¹, the κ₀/T of AuTe₂Se_4/3 in zero field is negligible. For s-wave nodeless superconductors, there are no fermionic quasiparticles to conduct heat in the zero-temperature limit, since all electrons form Cooper pairs.^[24,25] Therefore there is no residual linear term of κ₀/T, for examples in V₃Si and NbSe₂.^[25,27] However, for nodal superconductors, a substantial κ₀/T in zero field contributed by the nodal quasiparticles has been found.^[24] For example, κ₀/T of the overdoped d-wave cuprate superconductor Tl₂Ba₂CuO_6+δ (Tl-2201, T_c = 15 K) is 1.41 mW · K⁻² · cm⁻¹.^[28] For the p-wave superconductor Sr₂RuO₄ (T_c = 1.5 K), κ₀/T = 17 mW · K⁻² · cm⁻¹ was reported.^[29] Therefore, the negligible κ₀/T of AuTe₂Se_4/3 strongly suggests nodeless superconducting gap in it.

In Fig. 3(b), between H = 0 T and 0.5 T, we fit all the curves to obtain the κ₀/T for each magnetic field. Since κ/T starts to saturate at 0.40 T, we take H = 0.40 T as the bulk H_c2(0) of AuTe₂Se_4/3. This value is slightly lower than that determined by resistivity measurements in Fig. 2. Note that the choice of a slightly different H_c2(0) does not affect our discussion on the field dependence of κ₀/T below. The obtained κ₀/T = 0.28 ± 0.004 mW · K⁻² · cm⁻¹ in H_c2(0) = 0.40 T meets the normal-state Wiedemann–Franz law expectation L₀/ρ₀ = 0.28 mW · K⁻² · cm⁻¹ very well, with the Lorenz number L₀ = 2.45 × 10⁻⁸ W · Ω · K⁻² and ρ₀(0.4 T) = 87.3 μΩ · cm. The verification of the Wiedemann–Franz law in the normal state demonstrates that our thermal conductivity measurements are reliable.

The field dependence of κ₀/T can provide further information on the superconducting gap structure.^[24] In Fig. 4, we plot the normalized κ₀(H)/T as a function of H/H_c2 for AuTe₂Se_4/3. For comparison, similar data of the clean s-wave superconductor Nb,^[30] the dirty s-wave superconducting alloy InBi,^[31] the multiband s-wave superconductor NbSe₂,^[27] and iron-based superconductor BaFe_1.9Ni_0.1As₂,^[32] and an overdoped d-wave cuprate superconductor Tl-2201^[28] are also plotted. For a single band s-wave superconductor Nb,^[30] the κ(H)/T changes little even up to 40% H_c2, while for the nodal superconductor Tl-2201, a small field can yield a quick growth in the quasiparticle density of states (DOS) due to the Volovik effect, and the low-field κ₀(H)/T shows a roughly dependence of .^[28] In the case of NbSe₂, the distinct κ₀(H)/T behavior was well explained by multiple superconducting gaps with different magnitudes.^[27]

Fig. 4.

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	Fig. 4. (color online) Normalized residual linear term κ0/T of AuTe2Se4/3 as a function of H/Hc2. For comparison, similar data are shown for the clean s-wave superconductor Nb,[30] the dirty s-wave superconducting alloy InBi,[31] the multiband s-wave superconductor NbSe2,[27] and iron-based superconductor BaFe1.9Ni0.1As2,[32] and an overdoped d-wave cuprate superconductor Tl-2201.[28]

From Fig. 4, the field dependence of κ₀(H)/T for AuTe₂Se_4/3 grows slightly faster than that of the dirty s-wave superconductor InBi and lower than that of the multiband s-wave superconductor NbSe₂. In fact, it almost overlaps that of BaFe_1.9Ni_0.1As₂. In NbSe₂, the gap on the Γ band is approximately one-third of the gap on the other two Fermi surfaces, and the magnetic field first suppresses the superconductivity on the Fermi surface with smaller gap,^[27] thus κ₀/T(H) increases much rapid at low field. The optimally electron-doped BaFe_1.9Ni_0.1As₂ has a slower field dependence of κ₀(H)/T than NbSe₂.^[32] Previously, it was explained by using the multiple superconducting gaps of its sister compound, the optimally electron-doped BaFe_1.85Co_0.15As₂, which has the average gap values Δ(0) = 6.6 meV and 5.0 meV for hole and electron pockets, respectively.^[33] This ratio of different gap magnitudes R = 6.6/5.0 = 1.3 is much smaller than that of NbSe₂ (R ≈ 3). Due to the similarity between the κ₀(H)/T of AuTe₂Se_4/3 and BaFe_1.9Ni_0.1As₂, AuTe₂Se_4/3 is likely also a multiband s-wave superconductor. Indeed, electronic structure calculations of AuTe₂Se_4/3 show there are four bands crossing the Fermi level along different directions in Brillouin zone, including three 3D bands and one quasi-1D band.^[16] In this sense, the ratio of the magnitudes for different superconducting gaps in AuTe₂Se_4/3 should be around 1.3 as well.

So far, among all multi-band s-wave superconductors, only the iron-based superconductors are unconventional with the possible s_±-wave pairing.^[34] Therefore, the multiple s-wave gap we observe in AuTe₂Se_4/3 indicates conventional superconductivity in it. This is reasonable, since unconventional superconductivity usually involves magnetism,^[35] while AuTe₂Se_4/3 is nonmagnetic. As a new vdW quasi-2D superconductor, it will be interesting to fabricate AuTe₂Se_4/3 thin flakes to investigate whether its superconductivity can be enhanced by gating techniques, like in other quasi-2D compounds.^[8,36–41]

In summary, the superconducting gap structure of the new gold-based quasi-2D superconductor AuTe₂Se_4/3 has been studied by the ultra-low temperature thermal conductivity measurements. A negligible κ₀/T in zero field and the field dependence of κ₀/T suggest multiple nodeless superconducting gaps. This is consistent with the band structure calculations which shows four Fermi surfaces in AuTe₂Se_4/3. To confirm those multiple gaps, further experiments like angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling microscope (STM) are needed.