Nodes in the gap structure of the iron arsenide superconductor Ba(Fe1xCox)2As2 from c-axis heat transport measurements

Phys. Rev. B 82, 064501
(Color online) Second model for the gap structure. Same as in Fig. 12, except that the modulation of the superconducting gap Δ(k) is now a function of the azimuthal angle ϕ in the basal plane.
(Color online) First model for the gap structure. Simplified two-band model of the Fermi surface of Co-Ba122 (blue solid line), shown in the ac plane, with kzc in the vertical direction. Although in reality the Fermi surface of Co-Ba122 consists of at least four sheets, in our model we reduce it to two sheets: a sheet with strong 3D character (FS #1; top left) and a sheet with quasi-2D character (FS #2; top right). The superconducting gap Δ(k) on both Fermi surfaces (red dashed line) varies strongly as a function of kz, as shown on the bottom for three representative concentrations x from xs upward. On the 3D Fermi surface (#1), the gap modulation is such that it extends to negative values, producing nodes (black circles) at certain points. On the 2D Fermi surface (#2), the gap modulation is strong enough to cause a deep minimum (where Δ=Δmin), but not nodes. At x=xs, the gap minima are shallow and there are no nodes on either Fermi surface. With increasing x (decreasing Tc), the modulation increases, the minima deepen and the nodes appear. A further increase in x (beyond the maximal concentration in this study) could eventually also yield nodes on FS #2.(Color online) First model for the gap structure. Simplified two-band model of the Fermi surface of Co-Ba122 (blue solid line), shown in the ac plane, with kzc in the vertical direction. Although in reality the Fermi surface of Co-Ba122 consists of... Show more
(Color online) Bottom panel: residual linear term κ0/T of Co-Ba122 normalized by the normal-state value κN/T as a function of magnetic field H, plotted as (κ0/T)/(κN/T)κ0/κN vs H/Hc2 for three representative Co concentrations, as indicated: underdoped (x=0.042; red), slightly overdoped (x=0.074; black), and strongly overdoped (x=0.127; blue). κN/T is obtained from the Wiedemann-Franz law (see text and Tables ), except for the samples with x=0.127 where we use the value of κ0/T measured at H=15T, since Hc2=15T at that concentration. Full circles are for a heat current along the c axis (Jc; data from three A samples—see Table ). Empty circles are for a heat current along the a axis (Ja; data from three A samples—see Table ). The vertical dashed line marks H=Hc2/4; the value of κ0/κN at Hc2/4 is plotted in the top panel of Fig. 9, for all x. We also reproduce corresponding data for the d-wave superconductor Tl-2201 (from Ref. 59), the isotropic s-wave superconductor Nb, and the multiband s-wave superconductor NbSe2 (from Ref. 63). Top panel: anisotropy of the normalized residual linear term κ0/κN at x=0.127. The red dashed line at Hc2/10 marks roughly the field beyond which κ0/κN becomes isotropic.(Color online) Bottom panel: residual linear term κ0/T of Co-Ba122 normalized by the normal-state value κN/T as a function of magnetic field H, plotted as (κ0/T)/(κN/T)κ0/κN vs H/Hc2 for three representative Co concentrations, as indicated: underdop... Show more
(Color online) Comparison of heat transport by nodal quasiparticles and the jump in heat capacity at the superconducting transition, in Co-Ba122 as a function of Co concentration x. Heat transport is measured as the zero-field residual linear term in the thermal conductivity along the c axis, κc0/T, normalized by the corresponding normal-state conductivity at T0, κcN/T, multiplied by Tc. The heat-capacity jump ΔC is divided by Tc (from Ref. 61). The vertical dashed line marks the location of xs. Other lines are a guide to the eyes.(Color online) Comparison of heat transport by nodal quasiparticles and the jump in heat capacity at the superconducting transition, in Co-Ba122 as a function of Co concentration x. Heat transport is measured as the zero-field residual linear term in... Show more
(Color online) Field dependence of the residual linear term κa0/T in the a-axis thermal conductivity of our nine a-axis single crystals of Co-Ba122, with x as indicated. Underdoped compositions are shown of the left, overdoped compositions on the right. For three concentrations, two different crystals with nominally the same x value, labeled A (circles) and B (triangles), were measured (see Table ).(Color online) Field dependence of the residual linear term κa0/T in the a-axis thermal conductivity of our nine a-axis single crystals of Co-Ba122, with x as indicated. Underdoped compositions are shown of the left, overdoped compositions on the rig... Show more
(Color online) Field dependence of the residual linear term κc0/T in the c-axis thermal conductivity of our twelve c-axis single crystals of Co-Ba122 with x as indicated. Underdoped compositions are shown on the left, overdoped compositions on the right. For five concentrations, two different crystals with nominally the same x value, labeled A (circles) and B (triangles), were measured (see Table ).(Color online) Field dependence of the residual linear term κc0/T in the c-axis thermal conductivity of our twelve c-axis single crystals of Co-Ba122 with x as indicated. Underdoped compositions are shown on the left, overdoped compositions on the ri... Show more
(Color online) Temperature dependence of the thermal conductivity κ, plotted as κ/T vs T2, measured along the c axis in a Co-Ba122 sample with x=0.127, using the two-probe technique, for three values of the applied field: H=0.0, 0.05, and 0.10 T. At H=0, the total thermal resistance of sample plus contacts is dominated by the very high thermal resistance of the two superconducting tin contacts. At H=0.05T, tin is no longer superconducting and the thermal resistance of the two contacts has become negligible compared to the sample resistance. In this case, extrapolation of κ/T to T=0 gives us almost exactly the residual linear term in the sample’s thermal conductivity. Increasing H slightly beyond 0.05 T, for example to 0.10 T, leads to no further change in the data. This shows that measurements in H=0.05T reveal the intrinsic zero-field behavior of the sample. The inset shows the arrangement of the tin (Sn) contacts and the silver (Ag) wires on the sample.(Color online) Temperature dependence of the thermal conductivity κ, plotted as κ/T vs T2, measured along the c axis in a Co-Ba122 sample with x=0.127, using the two-probe technique, for three values of the applied field: H=0.0, 0.05, and 0.10 T. At ... Show more
(Color online) Residual linear term κ0/T of Co-Ba122 normalized by the normal-state value κN/T as a function of Co concentration x, at H=0 (lower panel) and H=Hc2/4 (upper panel). Full blue symbols are for a heat current along the c axis (Jc; circles for A samples, triangles for B samples in Table ). Empty red symbols are for a heat current along the a axis (Ja; circles for A samples, triangles for B samples in Table ). The white interval between the two vertical gray bands at x<0.032 and x>0.17 is the region of superconductivity in the phase diagram (Fig. 1). The vertical dashed line at x=0.06 marks the approximate location of the critical doping xs where the structure at T=0 goes from orthorhombic (below) to tetragonal (above) (Ref. 9) (see Fig. 1). Lines through the data points are a guide to the eyes. Error bars on the H=0 data are shown for the A samples (circles).(Color online) Residual linear term κ0/T of Co-Ba122 normalized by the normal-state value κN/T as a function of Co concentration x, at H=0 (lower panel) and H=Hc2/4 (upper panel). Full blue symbols are for a heat current along the c axis (Jc; circle... Show more
(Color online) Temperature dependence of the a-axis thermal conductivity κa, plotted as κa/T vs T, for six samples of Ba(Fe1xCox)2As2, with x as indicated, in a magnetic field H=0, 4, and 15 T (data taken at other fields are not shown for clarity). These are the six samples labeled A in Table . The lines are a power-law fit to the data below T=0.3K, namely, κ/T=a+bTα1. The fit is used to extrapolate κa/T to T=0 and thus obtain the residual linear term κa0/T. The power α is in the range from 2 to 2.5. The values of κa0/T are listed in Table for H=0 and plotted vs H in Fig. 8, for all nine a-axis samples. Solid black squares on the T=0 axis give the residual linear term in the normal-state thermal conductivity, κaN/T, obtained from the residual resistivity ρa0 of the sample via the Wiedemann-Franz law (see Table ).(Color online) Temperature dependence of the a-axis thermal conductivity κa, plotted as κa/T vs T, for six samples of Ba(Fe1xCox)2As2, with x as indicated, in a magnetic field H=0, 4, and 15 T (data taken at other fields are not shown for clarity). ... Show more
(Color online) Temperature dependence of the a-axis (in-plane) resistivity ρa(T) for the nine a-axis crystals of Ba(Fe1xCox)2As2 studied here with Co concentrations x as indicated. Top and bottom panels show underdoped and overdoped samples, respectively. Circles and triangles of the same color correspond to two different crystals at the same doping, respectively, labeled A and B (see Table ). The lines show how the data are extrapolated to T=0, to determine the value of the residual resistivity ρa0, given in Table .(Color online) Temperature dependence of the a-axis (in-plane) resistivity ρa(T) for the nine a-axis crystals of Ba(Fe1xCox)2As2 studied here with Co concentrations x as indicated. Top and bottom panels show underdoped and overdoped samples, respect... Show more
(Color online) Temperature dependence of the c-axis thermal conductivity κc, plotted as κc/T vs T2, for six samples of Ba(Fe1xCox)2As2, with x as indicated, in a magnetic field H=0.05, 4, and 15 T (data taken at other fields are not shown for clarity). These are six of the seven samples labeled A in Table . The lines are a linear fit to the data below T2=0.015K2, used to extract the residual linear term κc0/T as the extrapolation of κc/T to T=0. The values of κc0/T are listed in Table for H=0 and plotted vs H in Fig. 6, for all twelve c-axis samples. Solid black squares on the T=0 axis (dashed line) give the residual linear term in the normal-state thermal conductivity, κcN/T, obtained from the residual resistivity ρc0 of the sample via the Wiedemann-Franz law (see Table ).(Color online) Temperature dependence of the c-axis thermal conductivity κc, plotted as κc/T vs T2, for six samples of Ba(Fe1xCox)2As2, with x as indicated, in a magnetic field H=0.05, 4, and 15 T (data taken at other fields are not shown for clarit... Show more
(Color online) Temperature dependence of the c-axis resistivity ρc(T) for the twelve c-axis crystals of Ba(Fe1xCox)2As2 studied here with Co concentrations x as indicated. Top and bottom panels show underdoped and overdoped samples, respectively. Circles and triangles of the same color correspond to two different crystals at the same doping, respectively, labeled A and B (see Table ). The lines show how the data is extrapolated to T=0, to determine the value of the residual resistivity ρc0, given in Table .(Color online) Temperature dependence of the c-axis resistivity ρc(T) for the twelve c-axis crystals of Ba(Fe1xCox)2As2 studied here with Co concentrations x as indicated. Top and bottom panels show underdoped and overdoped samples, respectively. Ci... Show more
(Color online) Phase diagram of Ba(Fe1xCox)2As2 as a function of Co concentration x, showing the orthorhombic phase below Ts (blue), the antiferromagnetic (AF) phase below TN (green), and the superconducting phase below Tc (small black dots), as determined from resistivity, magnetization, and heat-capacity data (Ref. 8) and from x-ray (Ref. 9) and neutron (Ref. 10) data. Red circles mark the Tc value of seven samples measured in the present study (the c-axis samples labeled A in Table ), indicating the range of concentrations covered. The vertical thick line at x=0.06 marks the approximate location of the critical concentration xs where at T=0 the system goes from orthorhombic (Ortho) in the underdoped region (to the left) to tetragonal (Tetra) in the overdoped region (to the right) (Ref. 9). At T=0, the AF phase also ends close to x=0.06 (Ref. 10).(Color online) Phase diagram of Ba(Fe1xCox)2As2 as a function of Co concentration x, showing the orthorhombic phase below Ts (blue), the antiferromagnetic (AF) phase below TN (green), and the superconducting phase below Tc (small black dots), as det... Show more
I. INTRODUCTION

The discovery of superconductivity in iron arsenides, 1 with transition temperatures exceeding 50 K, 2 breaks the monopoly of cuprates as the only family of high-temperature superconductors and revives the question of the pairing mechanism. Because the mechanism is intimately related to the symmetry of the order parameter, which is, in turn, related to the k dependence of the gap function Δ(k), it is important to determine the gap structure in the iron-based superconductors, just as it was crucial to establish the d-wave symmetry of the gap in cuprate superconductors. The gap structure of iron-based superconductors has been the subject of numerous studies (for recent reviews, see Refs. 3,4). Here, we focus on the material BaFe2As2, in which superconductivity can be induced either by applying pressure 5 or by various chemical substitutions, such as K for Ba (K-Ba122) (Ref. 6) or Co for Fe (Co-Ba122). 7 In the case of Co-Ba122, single crystals have been grown with compositions that cover the entire superconducting phase (see Fig. 1). 8–10

Two sets of experiments on doped BaFe2As2 appear to give contradictory information. On the one hand, angle-resolved photoemission spectroscopy (ARPES) detects a nodeless, isotropic superconducting gap on all sheets of the Fermi surface in K-Ba122 (Ref. 11) and in optimally doped Co-Ba122, 12 and tunneling studies in K-Ba122 detect two full superconducting gaps. 13 The magnitude of the gaps in ARPES is largest on Fermi surfaces where a density-wave gap develops in the parent compounds. 14 This is taken as evidence for an s± pairing state driven by antiferromagnetic (AF) correlations.4,15–17 On the other hand, the penetration depth in Co-Ba122,18,19 the spin-lattice-relaxation rate in K-Ba122, 20 and the in-plane thermal conductivity in K-Ba122 (Ref. 21) and Co-Ba122,22,23 for example, are inconsistent with a gap that is large everywhere on the Fermi surface. Note, however, that the evidence for deep minima in the gap is particularly clear in the overdoped regime,20,22,24 a regime which has not so far been probed by either ARPES or tunneling.

Another possible explanation for the apparent discrepancy between the two sets of experimental results is a different sensitivity to the c-axis component (kz) of the quasiparticle k vector, taking into account the three-dimensional (3D) character of the Fermi surface.7,25–31 Nodes along the c axis were suggested theoretically to explain the discrepancy between ARPES, penetration depth and NMR studies.32,33 A variation in the gap magnitude as a function of kz was suggested in experimental studies of the neutron resonances in optimally doped Ni-Ba122. 34 It was also invoked to explain the temperature dependence of the penetration depth in Co-Ba122 (Ref. 19) and its a-c anisotropy in Ni-Ba122. 24 Clearly, it has become important to resolve the 3D structure of the superconducting-gap function in doped BaFe2As2.

Heat transport measured at very low temperatures is one of the few directional bulk probes of the gap structure. The existence of a finite residual linear term κ0/T in the thermal conductivity κ(T) as T0 is unambiguous evidence for the presence of nodes in the gap, 35–39 and thus by measuring κ(T) as a function of direction in the crystal, one can locate the position of nodes on the Fermi surface.35,36,40,41 Here, we report measurements of heat transport in Ba(Fe1xCox)2As2 for a current direction both parallel and perpendicular to the c axis of the tetragonal (Tetra) [or orthorhombic (Ortho)] crystal structure. Our main finding is a sizable residual linear term κ0/T for a current along the c axis and a negligible one for a current perpendicular to it. This implies the presence of nodes in the gap in regions of the Fermi surface that dominate the c-axis conduction and contribute little to in-plane conduction. Our study shows that the gap structure of Co-Ba122 depends on the 3D character of the Fermi surface in a way that varies strongly with x.

II. EXPERIMENTAL
A. Samples

Single crystals of Ba(Fe1xCox)2As2 were grown from FeAs: CoAs flux, as described elsewhere. 8 The doping level in the crystals was determined by wavelength dispersive electron-probe microanalysis, which gave a Co concentration, x, roughly 0.7 times the flux-load composition (or nominal content). We studied seven compositions: underdoped, with x=0.038, 0.042, and 0.048; overdoped, with x=0.074, 0.108, 0.114, and 0.127. The uncertainty in the determination of x is ±0.001. In this paper, “underdoped” and “overdoped” refer to concentrations, respectively, below and above the critical concentration xs0.06 at which the system at T=0 goes from orthorhombic (below) to tetragonal (above). 9 The Tc value for each composition is shown on the phase diagram in Fig. 1. A total of twelve c-axis and nine a-axis samples were studied; their characteristics are listed in Tables I and II, respectively. Three of the a-axis samples were the subject of a previous study (0.074-B, 0.108-A, and 0.114-B). 22

TABLE I.

Properties of the twelve c-axis samples of Co-Ba122 used in this study. x is the Co concentration measured by wavelength dispersive microprobe analysis. The superconducting transition temperature Tc is the temperature at which the resistivity goes to zero. The values of the upper critical field Hc2 (along the c-axis) needed to suppress superconductivity in Co-Ba122 at T0 are taken from Refs. 8,42. [ Hc2(T) is defined as the end, or “offset,” of the superconducting drop in ρ(T) vs H]. The residual resistivity ρc0 is obtained by a smooth extrapolation of the ρc(T) data to T=0, as shown in Fig. 3. The normal-state residual linear term in the thermal conductivity, κcN/T, is obtained from the Wiedemann-Franz law applied to ρc0 (see text). The zero-field residual linear term, κc0/T, is obtained by extrapolating to T=0 the zero-field thermal conductivity κc with a linear fit to κc/T vs T2, as shown in Fig. 5. κc0/T is also expressed as a fraction of the normal-state κcN/T, denoted κ0/κN(κ0/T)/(κN/T).

x Sample Tc (K) Hc2 (T) ρc0 (μΩcm) κcN/T (μW/K2cm) κc0/T (μW/K2cm) κ 0 / κ N
0.038 A 9.7 30  1935  12.7 6.1 0.48
0.042 A 14.4 40  1980  12.4 2.3 0.19
0.042 B 13.7 40  2115  11.6 2.9 0.25
0.048 A 17.2 45  2535  9.7 0.6 0.06
0.048 B 17.2 45  3045  8.0 0.8 0.10
0.074 A 22.9 60  1030  23.8 0.9 0.04
0.074 B 24.1 60  1140  21.5 0.2 0.01
0.108 A 15.2 30  1560  15.7 2.3 0.15
0.108 B 14.6 30  1770  13.8 1.6 0.12
0.114 A 11.0 20  1415  17.3 3.8 0.22
0.127 A 8.4 15  1500  16.3 5.6 0.34
0.127 B 9.3 15  1130  21.7 6.8 0.31
TABLE II.

Properties of the nine a-axis samples of Co-Ba122 used in this study. x, Tc, Hc2, and κ0/κN are defined in Table I. The residual resistivity ρa0 is obtained by a smooth extrapolation of the ρa(T) data to T=0, as shown in Fig. 4. The normal-state residual linear term in the thermal conductivity, κaN/T, is obtained from the Wiedemann-Franz law applied to ρa0. The zero-field residual linear term, κa0/T, is obtained by extrapolating to T=0 the zero-field thermal conductivity κa with a power-law fit to κa/T (see text), as shown in Fig. 7. Note that the magnitude of κa0/T, whether positive or negative, is in all cases lower than the uncertainty in the extrapolation (see Fig. 9).

x Sample Tc (K) Hc2 (T) ρa0 (μΩcm) κaN/T (μW/K2cm) κa0/T (μW/K2cm) κ 0 / κ N
0.042 A 13.0 40  200  123  0.01
0.042 B 14.2 40  235  104 
0.048 A 16.7 45  150  163  0.01
0.074 A 22.2 60  62  395  −1 
0.074 B 22.2 60  82  299  0.01
0.108 A 14.8 30  59  415  −1 
0.114 A 10.8 20  59  415  −9  −0.02
0.114 B 10.2 20  56  438  −13  −0.03
0.127 A 8.2 15  48  510  17  0.03
B. Two-probe transport measurements

Thermal conductivity was measured in a standard one-heater-two-thermometer technique. 43 The magnetic field H was applied along the [001] or c-axis direction of the crystal structure, which is tetragonal for overdoped samples and orthorhombic for underdoped samples at low temperatures. The underdoped samples were studied in their twinned state, 44 averaging out any in-plane anisotropy.45,46 Data were taken on warming after having cooled in a constant field applied above Tc to ensure a homogeneous field distribution.

It is conventional to measure electrical and thermal resistances in a four-probe configuration to avoid the contribution of contact resistances. This is what was done for data taken with a current in the basal plane ( Ja, in the notation appropriate for the tetragonal phase), as described elsewhere.21,22 For a current along the c axis (Jc), however, the four-probe technique is difficult because of the strong tendency of iron-arsenide crystals to exfoliation, which makes it difficult to cut samples thick enough in the c direction to attach four contacts.29,30 Consequently, c-axis transport was measured using a two-probe technique, which is valid provided contact resistances are much smaller than the sample resistance.

Contacts to the c-axis samples were made using silver wires (of 50μm diameter), soldered to top and bottom surfaces of the sample with ultrapure tin (see inset of Fig. 2). The contact making and properties are described in detail in Ref. 47. In brief, these contacts are characterized by a surface area resistivity in the nanoOhm per square centimeter range, which, for a typical sample size, yields a contact resistance below 10μΩ. This is negligible compared to a typical sample resistance in the normal state, on the order of 10mΩ.

Because tin is a superconductor, the thermal resistance of the contacts at very low temperature is large. We therefore have to apply a small magnetic field to suppress the superconductivity of tin and make it a normal metal with the very low electrical and thermal resistance mentioned above. A field of 0.05 T is sufficient to do this. In Fig. 2, we compare data obtained with H=0.00, 0.05, and 0.10 T. The effect of switching off the contact resistance with the field is clear, and once tin has gone normal, the data is independent of a further small increase in H. We therefore regard the data taken at H=0.05T as representative of the zero-field state of the sample.

C. Electrical resistivity

In Fig. 3, we show the temperature dependence of the electrical resistivity ρc(T) of our c-axis samples, measured in a two-probe configuration. In all samples, the resistivity follows qualitatively the temperature dependence reported previously. 29 Data for the a-axis samples are shown in Fig. 4. A smooth extrapolation of ρ(T) to T=0 yields the residual resistivity ρ0 listed in Table I for c-axis samples and in Table II for a-axis samples. The uncertainty associated with the extrapolation of ρ(T) to T=0 is approximately ±5%. Due to the uncertainty in measuring the geometric factor, the absolute value of the resistivity has an error bar of approximately ±20% for a-axis samples and a factor of 2 uncertainty for c-axis samples. 29–31 The higher ρ0 values in the underdoped regime are due to a reconstruction of the Fermi surface in the antiferromagnetic phase. 48 The residual resistivity ρ0 is used to determine the normal-state thermal conductivity κN/T in the T=0 limit via the Wiedemann-Franz law, κN/T=L0/ρ0, where L0=2.45×108WΩ/K2. Because the same contacts are used for electrical and thermal measurements, the relative geometric-factor uncertainty between the measured κ and this electrically determined κN is minimal.

III. RESULTS
A. Heat transport in the c direction

The thermal conductivity of solids is the sum of electronic and phononic contributions: κ=κe+κp. In the T=0 limit, the electronic conductivity is linear in temperature: κeT. In practice, the way to extract κe is to extrapolate κ/T to T=0 and thus obtain the purely electronic residual linear term, κ0/T.40,43,49 If one can neglect electron-phonon scattering, as one usually can deep in the superconducting state, then the mean free path of phonons as T0 is controlled by the sample boundaries. If those boundaries are rough, the scattering is diffuse and the mean free path is constant, such that the phonon conductivity κpT3. (Phonons can also be scattered by twin boundaries and grain boundaries.) If the sample boundaries are smooth, specular reflection yields a temperature-dependent mean free path and κpTα, typically with 2<α<3.43,50

In Fig. 5, we show the thermal conductivity κc of our c-axis samples, plotted as κc/T vs T2 for magnetic fields from H=0.05 to 15 T. Below T0.15K, the curves are linear, consistent with diffuse phonon scattering on the sample boundaries of our c-axis samples, which are indeed characterized by rough side surfaces. We obtain κe/Tκ0/T by extrapolating κ/T to T=0 using a linear fit below T2=0.015K2. The error bar on this extrapolation is approximately ±0.5μW/K2cm for all c-axis samples. The value of κc0/T thus obtained is plotted as a function of field H in Fig. 6 for all twelve c-axis samples. For five concentrations, we have a pair of crystals with nominally the same Co concentration. As can be seen, the two curves in each pair are in good agreement with each other, well within the uncertainty in the geometric factor. The zero-field values are listed in Table I. They range from κc0/T<1μW/K2cm at x=0.048 and 0.074 to κc0/T6μW/K2cm at x=0.038 and 0.127.

The normal-state residual linear term κN/T was estimated using the values of ρ0 through application of the Wiedemann-Franz law. The value of κN/T is shown as a solid black square on the y axis of Fig. 5. For the most heavily overdoped samples, with x=0.127, a magnetic field of 15 T is sufficient to reach the normal state, where κ0/T saturates to its normal-state value κN/T. This allows us to check the Wiedemann-Franz law. For sample A, κc0/T=16.0±0.5μW/K2cm at H=15T while κN/T=16.3±0.8μW/K2cm; for sample B, κc0/T=20.0±0.5μW/K2cm at H=15T while κN/T=21.7±1.1μW/K2cm. Within error bars, associated with extrapolations to get κ0/T and ρ0, the Wiedemann-Franz law is satisfied in both samples.

B. Heat transport in the a direction

In Fig. 7, we show the thermal conductivity κa for six of our nine a-axis samples, plotted as κa/T vs T, for magnetic fields from H=0 to 15 T. Unlike in the c-axis samples, the phonon conductivity κp does not obey κp/TT2 as T0. Instead, it follows approximately a power law such that κp/TTα1 with 2.0<α<2.5. These values of α are typical of specular reflection off smooth mirrorlike surfaces.43,50 The cleaved surfaces of these Co-Ba122 crystals (normal to the c axis) are indeed mirrorlike. Previous measurements of in-plane heat transport on K-Ba122, 21 Co-Ba122, 23 and Ni-Ba122 (Ref. 51) have all obtained α<2.7.

As done previously for other a-axis samples, 22 we obtain the residual linear term κa0/T by fitting the data below T=0.3K to a power-law expression, κ/T=a+bTα1, where aκa0/T. The error bar on this extrapolation is approximately in the range ±1020μW/K2cm. (The uncertainty is an order of magnitude larger than for κc0/T because the phonon-related slope is an order of magnitude steeper.) As found previously over the concentration range 0.048x0.114, 22 we again find κa0/T0, within error bars, now over a wider range: 0.042x0.127. This is consistent with a separate report that κa0/T0 in Co-Ba122 at x=0.135. 23

Upon application of a magnetic field, κa0/T increases, as displayed in Fig. 8 for all nine a-axis samples. For three concentrations, we have a pair of a-axis crystals with nominally the same Co concentration. As can be seen, the two curves in each pair are in good agreement with each other, within the ±20% uncertainty in the geometric factor and the error bar on the extrapolations.

IV. DISCUSSION

The results of our study are summarized in Fig. 9, where the κ0/T values of all 21 samples are plotted vs x, normalized to their respective normal-state value κN/T.

A. Intrinsic nature of the residual linear term

An inhomogenous sample, in which a fraction of the volume is nonsuperconducting, would show a residual linear term that is not an intrinsic property of the superconducting state. Such sample inhomogeneity was probably responsible for the large residual linear term reported in the earliest measurements of a-axis heat transport on Ba(Fe1xCox)2As2, where κa0/κN0.6. 52 Subsequent studies gave negligible residual linear terms in κa.22,23 The fact that κa0/κN0 for all a-axis samples in our study rules out an extrinsic origin for the large values of κc0/κN observed along the c axis in samples grown in nominally identical conditions. Indeed, any nonsuperconducting region would make an isotropic contribution to the normalized conductivity κ0/κN, in stark contrast with the data of Fig. 9. Additional evidence against an extrinsic origin is the good reproducibility of κc0/κN for different samples at the same concentration and the smooth, monotonic dependence on x over a large number of different samples.

In recent measurements of the heat capacity of Ba(Fe1xCox)2As2, a sizable residual linear term γ0 was extracted after subtraction of a Schottky term. 53 In a sample with x=0.08 (the only one with a full Meissner volume), the zero-field value normalized to the normal state is γ0/γN0.2. 53 The authors suggest that this large γ0 could be due to nanoscale electronic inhomogeneity. In our four samples at x=0.074, the maximum value allowed by the error bars is κ0/κN0.03 for both directions of heat flow. This means that, in our samples at least, any inhomogeneity (nanoscale or otherwise) contributes not more than 3% of the normal-state conductivity. Note that both residual linear terms ( γ0 and κ0/T) could well be intrinsic, due to nodal quasiparticles. In a clean d-wave superconductor, γ0/γN can easily be an order of magnitude larger than κ0/κN. For an impurity scattering rate Γ0 such that Γ0=0.1kBTc, for example, γ0/γN0.3 (Ref. 54) and κ0/κN0.03. 55 We conclude that any inhomogeneity which may be present in our samples makes a negligible contribution to the large c-axis residual linear term that we measure.

B. Gap nodes
1. Zero magnetic field

Our central finding is the presence of a substantial residual linear term κ0/T in the thermal conductivity of Co-Ba122 in zero field for heat transport along the c axis. It implies the presence of nodes in the superconducting gap, such that Δ(k)=0 for some wave vectors k on the Fermi surface.35–37,40,56 Because heat conduction in a given direction is dominated by quasiparticles with k vectors along that direction,35,36 the fact that κ0/T is negligible when heat transport is along the a axis, at all x, implies that the nodes are located in regions of the Fermi surface that contribute strongly to c-axis conduction but very little to in-plane conduction. The anisotropy of κ0/T becomes pronounced as x moves away from the critical doping xs0.06, in either direction. For x>0.06, we see that the a/c anisotropy in κ0/κN is at least a factor 10 (see Fig. 9). Such a large anisotropy is not expected in a scenario of isotropic pair breaking. 57

At the highest doping studied here, x=0.127, κ0/κN=0.34±0.03 for Jc. (This is for sample A, which has the lowest Tc in the overdoped regime; see Table I.) This magnitude is typical of superconductors with a line of nodes in the gap. In the heavy-fermion superconductor CeIrIn5 with Tc=0.4K and Hc20.5T, κ0/κN0.2.41,58 In the ruthenate superconductor Sr2RuO4 with Tc=1.5K and Hc2=1.5T, κ0/κN0.10.3 (depending on sample purity). 39 In the overdoped cuprate Tl2Ba2CuO6δ (Tl-2201), a d-wave superconductor with Tc=15K and Hc27T, κ0/κN0.35. 59 In the latter case, because the order parameter is well known and the Fermi surface is very simple [a single two-dimensional (2D) cylinder], it was possible to show that the magnitude of κ0/T agrees quantitatively with the theoretical BCS expression for the residual linear term in a d-wave superconductor, 49 namely, κ0/T=(kB2/3d)(kFvF/S),36,37,40 where d is the interlayer separation, kF and vF are the Fermi wave vector and velocity at the node, respectively, and SδΔ/δk is the slope of the gap at the node. [For a d-wave gap with Δ(k)=Δ0cos(2ϕ), S=2Δ0.]

If the line of nodes in the gap is imposed by the symmetry of the order parameter, as in a d-wave state, then κ0/T is universal, i.e., independent of the impurity-scattering rate Γ0, in the clean limit Γ0Δ0.36,37 Such universal transport was demonstrated experimentally for CeIrIn5, 58 Sr2RuO4, 39 and the cuprates YBa2Cu3O7 (Ref. 38) and Bi2Sr2CaCu2O8. 60 As a fraction of the normal-state conductivity, one then gets (κ0/T)/(κN/T)κ0/κNΓ0/S. 36 However, if the nodes are not imposed by symmetry, but are “accidental,” as in an “extended- s-wave” state, they still cause a nonzero residual linear term, with κ0/T1/S, but κ0/T is no longer universal because S depends on the scattering rate Γ0. 56

In Fig. 9, we see that κc0/κcN exhibits a striking U-shaped dependence on Co concentration x with κc0/κcN0 as xxs. Just above xs, at x=0.074, κc0/T=0.2±0.5μW/K2cm. (This is for sample B, which is closest to xs, as it has the highest Tc; see Table I.) This is equal to zero within error bars, indicating that there are no nodes in the gap at this concentration, as also inferred from the field dependence (see below). If the nodes can be removed simply by changing x, then these nodes must be accidental, not imposed by symmetry.

Given that the change in κ0/κN with x on the overdoped side is due to a change in κ0/T and not a change in κN/T (since ρ0 is independent of x, within error bars), we attribute the dramatic rise in κ0/κN from x=0.074 to x=0.127 to a decrease in the slope S with increasing x. Part of this decrease must be due to a drop in the overall strength of superconductivity, as measured by the decreasing Tc. We can factor out that effect by multiplying κ0/κN by Tc, as shown in Fig. 10. We see that κ0/κN×Tc vs x is far from constant, as it would be if the decrease in Δ(k) vs x was uniform, independent of k. In a d-wave superconductor, for example, S would typically scale with the gap maximum Δ0, which itself would scale with Tc, giving a constant product κ0/κN×Tc (for a constant Γ0). By contrast, in Co-Ba122 the slope of the gap at the nodes decreases faster than that part of the gap structure which controls Tc. In other words, Δ(k) must be acquiring a stronger and stronger k dependence, or modulation, with increasing x.

In the underdoped regime, for samples with x=0.048 and lower, the metal is antiferromagnetic 10 and its Fermi surface is reconstructed by the antiferromagnetic order. Nevertheless, a residual linear term κc0/T is still observed at H=0 (see Fig. 9). At x=0.038, it is even larger than at x=0.127, namely, κ0/κN=0.48±0.04 (see Table I). This implies that nodes are present in the superconducting gap inside the region of coexisting antiferromagnetic order. The fact that κ0/T is again strongly anisotropic (see Fig. 9) means that those nodes are still located in regions of the Fermi surface that contribute strongly to c-axis conduction and little to a-axis conduction. The fact that the nodes survive the Fermi-surface reconstruction is consistent with their location in regions with strong 3D character, since the spin-density wave gaps the nested portions of the Fermi surface, which are typically those with strong 2D character. (It should be emphasized that the mechanisms responsible for the drop in Tc and the rise in κ0/κN are likely to be different above and below optimal doping.)

2. Field dependence

The effect of a magnetic field on κ0/T reveals how easy it is to excite quasiparticles at T=0.40,56,62 For a gap with nodes, the rise in κ0/T with H is very fast because delocalized quasiparticles exist outside the vortices, 62 as shown for the d-wave superconductor Tl-2201 in Fig. 11. For a full gap without nodes or deep minima, such as in the s-wave superconductor Nb, the rise in κ0/T vs H is exponentially slow (see Fig. 11), because it relies on tunneling between quasiparticle states localized on adjacent vortices. For Co-Ba122 at x=0.127, κc0/κN is seen to track the d-wave data all the way from H=0 to H=Hc2. This nicely confirms the presence of nodes in the gap structure of overdoped Co-Ba122 that dominate the transport along the c axis.

By contrast, at x=0.074, the initial rise in κc0/T vs H has the positive (upward) curvature typical of a nodeless gap for both samples A and B (see Fig. 6). The rise at low H is faster than in a simple s-wave superconductor like Nb (Fig. 11), either because of a k dependence of the gap or because of a multiband variation in the gap amplitude, or both. A multiband variation is what causes the fast initial rise in κ0/T vs H (with positive curvature) in NbSe2 (Ref. 63) (see Fig. 11). This H dependence strongly suggests that there are no nodes in the gap of Co-Ba122 at x=0.074, as inferred above from the negligible value of κa0/T.

C. Gap minima

We saw that nodes in the gap have two general and related signatures in the thermal conductivity: 40 (1) a finite residual linear term κ0/T in zero field and (2) a fast initial rise in κ0/T with H. Both signatures are clearly observed in Co-Ba122 at x=0.127 for Jc. For Ja, however, the situation is quite different. Indeed, κa0/T is negligible at H=0, for all x, as also found in previous measurements of κa on underdoped K-Ba122, 21 optimally doped Ni-Ba122, 51 and overdoped Co-Ba122. 23

Consequently, the fast initial rise in κ0/T with H for Ja, seen in Fig. 11, is not due to nodes but rather to the presence of deep minima in the gap, in regions of the Fermi surface that contribute significantly to in-plane conduction, as previously reported. 22 In the top panel of Fig. 9, we show the normalized residual linear term κ0/κN measured at H=Hc2/4. We see that in the overdoped regime the κ0/κN values are the same for both current directions, at all x. In other words, whereas κ0/κN is very anisotropic at H=0, it is essentially isotropic at H>Hc2/10, as shown for x=0.127 in the top panel of Fig. 11. But quasiparticle transport for Jc is due to nodal excitations whereas quasiparticle transport for Ja comes from field-induced excitations across a minimum gap. In a single-band model, say, with a single ellipsoidal Fermi surface, this contrast between zero-field anisotropy and finite-field isotropy can only be described by invoking two unrelated features in the gap structure Δ(k): nodes along the c axis and deep minima in the basal plane. However, the fact that κ0/κN remains isotropic at all x (for H=Hc2/4) strongly suggests that nodes and minima are, in fact, intimately related. We therefore propose that they both come from the same tendency of the gap function Δ(k) to develop a strong modulation as a function of k, which causes a deep minimum on one Fermi surface and an even deeper minimum on another Fermi surface, where the gap would actually go to (or through) zero. In other words, instead of invoking two unrelated features of the gap structure on a single Fermi surface, we invoke a single property of the gap structure which leads to two related manifestations on separate Fermi surfaces.

D. Two simple models for the gap structure

For the purpose of illustration, we consider a simplified two-band model for the Fermi surface, whereby one surface has strong 3D character and the other has quasi-2D character, as sketched in Fig. 12. The 3D Fermi surface can either be open along the c axis, as drawn in Fig. 12 and suggested by some ARPES data, 25 or closed, as suggested by some band-structure calculations. 64 The 3D Fermi surface is responsible for most of the c-axis conduction and the 2D surface for most of the a-axis conduction (recall that in the T=0 limit κaN/κcN=ρcN/ρaN20). Note that in reality the Fermi surface of Co-Ba122 contains at least four separate sheets;33,64 our model requires that at least one of these has strong 3D character and it treats all others in terms of a single Fermi surface, the second quasi-2D sheet. We then propose that the gap Δ(k) varies strongly as a function of k, on both Fermi surface sheets. There are two basic scenarios: a gap modulation as a function of kz, illustrated in Fig. 12, or a gap modulation as a function of the azimuthal angle ϕ in the basal plane, illustrated in Fig. 13. The strong modulation extends to negative values on the 3D Fermi surface, thereby producing nodes where Δ(k)=0, whereas it only produces a deep minimum (where Δ=Δmin) on the 2D Fermi surface (at least in the range of concentrations covered here). In the first scenario (Fig. 12), the lines of nodes are horizontal circular loops in a plane normal to the c axis; in the second scenario (Fig. 13), they are vertical lines along the c axis.

Both versions of the model explain the isotropy at Hc2/4 and the anisotropy at H=0. The isotropy of κ0/κN follows fundamentally from having a similar k modulation of the gap on both Fermi surfaces. When the field is large enough to excite quasiparticles across the minimum gap on the 2D Fermi surface, quasiparticle transport from both Fermi surfaces will be similar, explaining the rapid and isotropic rise in κ0/T with H. By contrast, at H=0 no quasiparticles are excited on the 2D Fermi surface at T=0 (since kBTΔmin), whereas nodal quasiparticles are always present on the 3D surface. This explains the large anisotropy of κ0/κN at H=0. Note that this anisotropy is not governed by the anisotropy of the gap itself, i.e., by the direction of the gap modulation, but rather by the fact that the nodes lie on the 3D Fermi surface. (Whether horizontal or vertical line nodes are more consistent with our data depends on details of the real Fermi surface of Co-Ba122.) The nodal quasiparticles on the 3D sheet must also contribute to a-axis conduction. Assuming that for the 3D Fermi surface κa0/Tκc0/T at H=0, we should detect a residual linear term κa0/T6μW/K2cm in the a-axis sample with x=0.127, for example. This is indeed consistent, within error bars, with the value we extrapolate for the a-axis data at x=0.127 (Fig. 7), namely, κa0/T=17±20μW/K2cm (Table II).

In both versions of our model for the gap structure, the U-shaped x dependence of κ0/κN is attributed to an increase in the modulation of the gap as x moves away from xs, as illustrated in Figs. 12 and 13. The fact that the U-shaped curves in Fig. 9 have their minimum where the (inverted U-shaped) Tc vs x curve has its maximum points to a reverse correlation between Tc and gap modulation. Modulation is a sign of weakness. The presence of nodes in the gap may then be an indicator that pairing conditions are less than optimal.

It is possible that at high enough x in the overdoped regime Δmin, the minimum value of the gap on the quasi-2D Fermi surface goes to zero so that nodes appear on that Fermi surface as well. This would immediately cause κa0/T to become sizable. It is conceivable that the large value of κa0/T measured in undoped KFe2As2, 65 which can be viewed as the strong doping limit of K-Ba122, is the result of a gap modulation so strong that it goes to (through) zero on all Fermi surfaces.

A pronounced modulation of Δ(k) should manifest itself in a number of physical properties. For example, in an s-wave superconductor, a variation in the gap magnitude over the Fermi surface, whether from band to band as in MgB2 (Ref. 66) or from k dependence (anisotropy) as in Zn, 67 leads to a suppressed ratio of specific-heat jump ΔC at the transition to Tc. The pronounced gap modulation and anisotropy revealed by the thermal conductivity could therefore account for the dramatic variation in ΔC/Tc measured in Co-Ba122 vs x, 61 reproduced in Fig. 10. ΔC/Tc is seen to be maximal where κ0/κN is minimal, i.e., where the gap modulation is weakest, and it drops just as rapidly with a change in x as κ0/κN rises.

E. Theoretical calculations

The two-band picture suggested by our thermal conductivity data is reminiscent of the proximity scenario proposed for the three-band quasi-2D p-wave superconductor Sr2RuO4, 68 where superconductivity originates on one band, the most 2D one, and is induced by proximity on the other two bands. This k-space promixity effect is such that a kz modulation of the induced gap produces horizontal line nodes on the latter two Fermi surfaces, 68 in analogy with the horizontal-line scenario of Fig. 12. A proximity scenario of this sort was, in fact, proposed for the pnictides, 32 predicting c-axis nodes in the superconducting gap. The effect on the superconducting gap structure of including the kz dispersion of the Fermi surface in BaFe2As2 was recently calculated within a spin-fluctuation pairing mechanism on a 3D multiorbital Fermi surface. 33 A strong modulation of the gap Δ(k) as a function of both kz and ϕ is obtained which can indeed, for some parameters, lead to accidental nodes.

The thermal conductivity of pnictides was calculated in a 2D two-band model for the case of an extended- s-wave gap (of A1g symmetry). 56 These calculations show that the presence of deep minima in the gap, in this case as a function of ϕ, can account for the rapid initial rise observed in κa0/T vs H, starting from κa0/T=0 at H=0. It seems clear that calculations for a gap whose deep minima occur instead as a function of kz would yield similar results. It will be interesting to see what calculations of the thermal conductivity give when applied to the 3D model of Ref. 33 or indeed to the simple two-band models proposed here (in Figs. 12 and 13).

V. CONCLUSIONS

In summary, our measurements of the thermal conductivity in the iron-arsenide superconductor Ba(Fe1xCox)2As2 show unambiguously that the gap Δ(k) has nodes. These nodes are present in both the overdoped and the underdoped regions of the phase diagram, implying that they survive the Fermi-surface reconstruction provoked by the antiferromagnetic order in the underdoped region. The nodes are located in regions of the Fermi surface that dominate c-axis conduction and contribute very little to in-plane conductivity. The fact that the strongly anisotropic quasiparticle transport at H=0 becomes isotropic in a magnetic field H=Hc2/4 shows that there must be a deep minimum in the gap in regions of the Fermi surface that dominate in-plane transport. These two features—nodes on 3D regions and minima on 2D regions of the Fermi surface—point to a strong modulation of the gap as a function of k. This modulation of Δ(k) would be present on all Fermi surfaces, but be most pronounced on that surface with strongest kz dispersion, where it has nodes. This suggests a close relation between the 3D character of the Fermi surface and gap modulation.

The anisotropy of κ shows a strong evolution with Co concentration x. At optimal doping, where Tc is maximal, there are no nodes and κ0/T has the anisotropy of the normal state. With increasing x, nodes appear and κ0/T acquires a strong anisotropy. We attribute this to an increase in the gap modulation with x, which may explain the strong decrease in the specific-heat jump at Tc (Ref. 61) and the change in the power-law temperature dependence of the penetration depth.18,19 The fact that nodes are located in regions that dominate c-axis conduction is consistent with the fact that the penetration depth along the c axis has a linear temperature dependence. 24

Horizontal line nodes in Co-Ba122, which would be the result of a strong modulation of the gap along kz rather than a strong in-plane angular dependence, would reconcile the isotropic azimuthal angular dependence of the gap seen by ARPES with the evidence of nodes or minima from thermal conductivity, NMR relaxation rate and penetration depth measurements in Co-Ba122 and other iron-based superconductors. A kz modulation of Δ(k) should be detectable by ARPES, especially in the overdoped regime where it would be strongest.

Because the nodes go away by tuning x toward optimal doping, we infer that they are “accidental,” i.e., not imposed by symmetry, and so consistent a priori with any superconducting order parameter, including the s± state. 15–17 Although accidental nodes are not a direct signature of the symmetry, the strong modulation of the gap nevertheless reflects an underlying k dependence of the pairing interaction, and as such the 3D character of the gap function Δ(k) is an important element in understanding what controls Tc in this family of superconductors.

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About the Authors

Image of J.-Ph. Reid
Image of M. A. Tanatar
Image of X. G. Luo
Image of H. Shakeripour
Image of N. Doiron-Leyraud
Image of N. Ni
Image of S. L. Bud’ko
Image of P. C. Canfield
Image of R. Prozorov
Image of Louis Taillefer

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