Nodeless superconducting gap in electron-doped BaFe$_{1.9}$Ni$_{0.1}$As$_2$ probed by quasiparticle heat transport

The in-plane thermal conductivity $\kappa$ of electron-doped iron-arsenide superconductor BaFe$_{1.9}$Ni$_{0.1}$As$_2$ ($T_c$ = 20.3 K) single crystal was measured down to 70 mK. In zero field, the absence of a residual linear term $\kappa_0/T$ at $ T \to 0$ is strong evidence for nodeless superconducting gap. In magnetic field, $\kappa_0/T$ shows a slow field dependence up to $H$ = 14.5 T ($\approx$ 30% $H_{c_2}$). This is consistent with the superconducting gap structure demonstrated by angle-resolved photoemission spectroscopy experiments in BaFe$_{1.85}$Co$_{0.15}$As$_2$ ($T_c$ = 25.5 K), where isotropic superconducting gaps with similar size on hole and electron pockets were observed.

The recent discovery of iron-based superconductors with T c as high as 55 K [1,2,3,4,5] has attracted great attention. As a second family of high temperature superconductors after cuprates, the pairing symmetry of its superconducting gap is one of the most important issue to address. Spin triplet pairing was first ruled out in BaFe 1.8 Co 0.2 As 2 (T c = 22 K) single crystal by the NMR Knight shift measurements [6]. This leaves three possible singlet paring candidates: conventional s, d, and s ± , a superconducting state with order parameters of opposite signs on the electron and hole pockets [7]. While Andreev spectroscopy [8], angle-resolved photoemission spectroscopy (ARPES) [9,10,11,12,13,14], and latest specific heat [15] experiments on FeAs-superconductors support full superconducting gaps without nodes, NMR data [16,17,18] and extensive penetration depth studies [19,20,21,22,23] reveal a contradictory picture of either nodeless or nodal superconductivity. Even if the nodeless superconducting gap is eventually confirmed, clear-cut experiments to distinguish s ± from conventional s-wave have to be done. Therefore, the paring symmetry in ironarsenide superconductors is still far from consensus.
Low-temperature thermal conductivity measurement is a powerful bulk tool to probe the superconducting gap structure [24]. For unconventional superconductors with nodes in the superconducting gap, like d-wave cuprates and p-wave ruthenate, the nodal quasiparticles will contribute a finite κ 0 /T in zero field [25,26]. So far, only one heat transport study was reported for FeAs-based superconductors [27]. For the hole-doped Ba 1−x K x Fe 2 As 2 (T c ≃ 30 K) single crystal, a negligible κ 0 /T was found in zero field, indicating a full superconducting gap. However, κ 0 /T increases rapidly with magnetic field even for H ≪ H c2 , which was inferred that the gap must be very small on some portion of the Fermi surface, whether from strong anisotropy or band dependence, or both. To clarify this important issue, more heat transport experiments on other FeAs-based superconductors are needed.
In this Letter, we probe the superconducting gap of electron-doped BaFe 1.9 Ni 0.1 As 2 by measuring the thermal conductivity κ of a single crystal with T c = 20.3 K down to 70 mK. In zero field, the residual linear term κ 0 /T is negligible, a clear indication that BaFe 1.9 Ni 0.1 As 2 has nodeless superconducting gap. In magnetic field, κ 0 /T (H) shows a slow field dependence, different from the case of hole-doped Ba 1−x K x Fe 2 As 2 . This difference is discussed on the base of superconducting gap structure in these two systems measured by ARPES.
Single crystals with nominal formula BaFe 1.9 Ni 0.1 As 2 were prepared by self flux method [28]. Energy Dispersive of X-ray (EDX) microanalysis show that the actual Ni content is 0.096, close to the nominal composition. The ac magnetization was measured in a Quantum Design Physical Property Measurement System (PPMS). The sample was cleaved to a rectangular shape of dimensions 1.5 × 0.88 mm 2 in the plane, with 55 µm thickness along the c-axis. Contacts were made directly on the fresh sample surfaces with silver paint, which were used for both resistivity and thermal conductivity measurements. The contacts are metallic with typical resistance 50 mΩ at 1.5 K. In-plane thermal conductivity was measured in a dilution refrigerator down to 70 mK, using a standard four-wire steady-state method with two RuO 2 chip thermometers, calibrated in situ against a reference RuO 2 thermometer. Magnetic fields were applied along the c-axis and perpendicular to the heat current. To ensure a homogeneous field distribution in the sample, all fields were applied at temperature above T c . Fig.  1a shows the in-plane resistivity of our BaFe 1.9 Ni 0.1 As 2 single crystal in zero field. The middle point of the resistive transition is at T c = 20.3 K, in good agreement with previous study [28]. The 10-90% width of the resistive transition is less than 0.3 K, indi- cating the high homogeneity of our crystal. The residual resistivity ρ 0 = 132 µΩ cm is extrapolated from the data above T c by using the Fermi liquid form ρ = ρ 0 + AT 2 . In Fig. 1b, the normalized ac magnetization also shows a sharp superconducting transition similar to Fig. 1a.
In Fig. 2, the temperature dependence of the in-plane thermal conductivity for BaFe 1.9 Ni 0.1 As 2 in zero field is plotted as κ/T vs T . Since both electrons and phonons contribute to the measured conductivity, we fit the data to κ/T = a + bT α−1 [29,30], where aT and bT α represent electronic and phonon contributions, respectively. For phonon scattering off the crystal bourdary at low temperature, one usually gets α = 3, but specular reflection of phonons at the smooth crystal surfaces can result in a lower power α < 3 [29,30]. For BaFe 1.9 Ni 0.1 As 2 , it is found that the data below 0.8 K can be well fitted (the solid line in Fig. 2) and gives κ 0 /T = -3 ± 2 µW K −2 cm −1 , with α = 2.02 ± 0.01.
Since the residual linear term κ 0 /T is within the experimental error bar ± 5 µW K −2 cm −1 [30], which is less than 3% of the normal-state value, the electronic contribution to the thermal conductivity is negligible in zero field. This is consistent with previous results on hole-doped Ba 1−x K x Fe 2 As 2 single crystals [27] and the low-T c superconductor BaNi 2 As 2 (T c = 0.7 K) [32], suggesting a nodeless (at least in ab-plane) superconducting gap. However, the power α = 2.02 of the phonon conductivity bT α is much lower than α = 2.65 found in Ba 1−x K x Fe 2 As 2 [27]. We note that in the parent compound BaFe 2 As 2 single crystal [31], the power α = 2.22 is more closer to our value. Whether specular reflections of the phonon boundary scattering [29,30] can give such a low α is not clear to us. In fact, phonons scattering off either electrons or grain boundaries will give α = 2 [33]. Therefore, more experimental results are needed to clarify the temperature dependence of phonon thermal conductivity in FeAs-compound single crystals. Fig. 3 shows the low-temperature thermal conductivity of BaFe 1.9 Ni 0.1 As 2 in magnetic fields applied along the c-axis (H = 0, 9, 13, and 14.5 T). The data of κ/T in high fields below 0.25 K manifests similar temperature dependence to the zero field data. We fit the H = 9, 13, and 14.5 T curves by using the same equation κ/T = a + bT α−1 , with fixed α = 2.02 due to the slighly increasing noise level of the in-field data. The solid lines are the fitting curves, which give κ 0 /T = 4, 15, and 20 µW K −2 cm −1 for H = 9, 13, and 14.5 T, respectively.
The upper critical field H c2 of BaFe 1.9 Ni 0.1 As 2 (T c = 20.3 K) single crystal has not been determined yet. For BaFe 1.8 Co 0.2 As 2 (T c = 22 K) single crystal, the H c2 was estimated ∼ 50 T [34]. Taking this value as the H c2 of our BaFe 1.9 Ni 0.1 As 2 sample, H = 14.5 T is just about 30% of H c2 .
In Fig. 4, the normalized κ 0 /T of BaFe 1.9 Ni 0.1 As 2 is plotted as a function of H/H c2 , together with the clean s-wave superconductor Nb [35], the dirty s-wave superconducting alloy InBi [36], the multi-band s-wave superconductor NbSe 2 [37], an overdoped sample of the d-wave superconductor Tl-2201 [25], and Ba 0.75 K 0.25 Fe 2 As 2 [27]. For a clean (like Nb) or dirty (like InBi) type-II s-wave superconductor with isotropic gap, κ 0 /T should grow exponentially with field (above H c1 ). This usually gives negligible κ 0 /T for field lower than H c2 /4, as seen in Fig. 4. For NbSe 2 , κ 0 /T increases much rapid at low field. This can be explained by its multi-gap structure, whereby the gap on the Γ band is approximately one third of the gap on the other two Fermi surfaces, and magnetic field will first suppresses the superconductivity on the Fermi surface with smaller gap (given that H c2 (0) ∝ ∆ 2 0 ) [37]. As seen in Fig. 4, the κ 0 /T (H) of BaFe 1.9 Ni 0.1 As 2 more likely follows the behavior of isotropic s-wave gap. This field dependence is different from that of the holedoped Ba 1−x K x Fe 2 As 2 (T c ≃ 30 K) sample [27], where κ 0 /T increases almost linearly with H up to 15 T. Such a rapid increase of κ 0 /T (H) in Ba 1−x K x Fe 2 As 2 has been interpreted as evidence for a k-dependent gap magnitude, coming from angle (i.e., anisotripic) or band (i.e., isotropic but with different magnetitudes on different bands) dependence, or both [27].
In order to explain this difference, let us examine the gap values on all Fermi surface (FS) sheets for both   [36], the multi-band s-wave superconductor NbSe2 [37], an overdoped sample of the d-wave superconductor Tl-2201 [25], and Ba0.75K0.25Fe2As2 [27] are also shown for comparison.
Since the doping level and T c of our BaFe 1.9 Ni 0.1 As 2 sample are close to those of BaFe 1.85 Co 0.15 As 2 , their superconducting gap structure should also be similar. Therefore, due to the similar sizes (6.6 vs 5.0 meV) of these isotropic superconducting gaps, the κ 0 /T (H) of our BaFe 1.9 Ni 0.1 As 2 sample behaves more like a conventional single-gap s-wave superconductor. For hole-doped Ba 0.6 K 0.4 Fe 2 As 2 , the sizes of these gaps are quiet different (12.5 vs 5.5 meV), which gives a ratio R = 12.5/5.5 = 2.3 [9,13]. Taking this ratio for the slightly underdoped Ba 1−x K x Fe 2 As 2 [27], it is smaller than that in NbSe 2 (R ≈ 3). This may explains the nearly linear increase of κ 0 /T (H) in Ba 1−x K x Fe 2 As 2 with the slope smaller than that in NbSe 2 [27], given H c2 (0) ∝ ∆ 2 0 and magnetic field first suppresses the superconductivity on the Fermi surface with smallest superconducting gap.
In summary, we have used low-temperature thermal conductivity to clearly demonstrate nodeless superconducting gap in electron-doped iron-arsenide superconductor BaFe 1.9 Ni 0.1 As 2 . Furthermore, the κ 0 /T (H) shows a slow H dependence at low field, different from the rapid, linear κ 0 /T (H) in hole-doped Ba 1−x K x Fe 2 As 2 . This dif-