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Novel anisotropy in the superconducting gap structure of Bi_{2}Sr_{2}CaCu_{2}O_{8+\delta} probed by quasiparticle heat transport

Since the nature of pairing interactions is manifested in the superconducting gap symmetry, the exact gap structure, particularly any deviation from the simple d_{x^2-y^2} symmetry, would help elucidating the pairing mechanism in high-T_c cuprates. Anisotropic heat transport measurement in Bi_{2}Sr_{2}CaCu_{2}O_{8+\delta} (Bi-2212) reveals that the quasiparticle populations are different for the two nodal directions and thus the gap structure must be uniquely anisotropic, suggesting that pairing is governed by interactions with a rather complicated anisotropy. Intriguingly, it is found that the"plateau"in the magnetic-field dependence of the thermal conductivity is observed only in the b-axis transport.

Since the nature of pairing interactions is manifested in the superconducting gap symmetry, the exact gap structure, particularly any deviation from the simple d x 2 −y 2 symmetry, would help elucidating the pairing mechanism in high-Tc cuprates. Anisotropic heat transport measurement in Bi2Sr2CaCu2O 8+δ reveals that the quasiparticle populations are different for the two nodal directions and thus the gap structure must be uniquely anisotropic, suggesting that pairing is governed by interactions with a rather complicated anisotropy. Intriguingly, it is found that the "plateau" in the magnetic-field dependence of the thermal conductivity is observed only in the b-axis transport. It has been established [1] that in the high-T c cuprates the superconducting gap ∆ has essentially the d x 2 −y 2 symmetry, ∆(k) = ∆ 0 cos 2φ, which is depicted in Fig. 1 (∆ 0 is the maximum gap and φ is the azimuthal angle). Although this information is very important, it does not help much in clarifying the high-T c mechanism, because all of the proposed theories of the high-T c superconductivity have successfully adopted the d-wave symmetry. It is thus important to obtain more detailed information on the gap structure, because any deviation from the simple d x 2 −y 2 symmetry would point to the role of a particular interaction in the paring. However, not much is known about the precise gap structure of the high-T c cuprates [2], partly because the experimental techniques to selectively probe the nodal structures are limited [3].
The most prominent feature of the d-wave gap is that it has four nodes where the gap magnitude vanishes. At finite temperature below T c , quasiparticles (QPs) are always thermally excited near these nodes from the superconducting condensate; how many QPs are excited depends on the "slope" of the gap at the node [4], because a steeper gap allows less phase space available below a given energy of k B T . Since these QPs carry heat (in contrast to the Cooper pairs that do not carry heat), one can investigate the gap structure of unconventional superconductors by measuring the QP heat transport in the superconducting state [5]. Moreover, because the QPs can only travel along the nodal directions [5], one can gain information about different nodes by tuning the direction of the heat current to the respective nodal directions; this makes it possible to study the detailed anisotropy of the gap structure in high-T c cuprates.
In recent years, the heat transport has been conveniently used to probe the gap structure of the high-T c cuprates. For example, it was reported [6,7] that the inplane thermal conductivity showed a four-fold-symmetric change when the heat current was fixed along an antinodal direction and an applied magnetic field was 360 • rotated in the CuO 2 plane, giving maxima for fields parallel to the nodal directions. This feature helped establishing the d x 2 −y 2 symmetry of the superconducting gap. The heat transport measurement was also employed in an intriguing proposal of a field-induced transition in the gap structure; namely, Krishana et al. reported [8] that at low temperatures the magnetic-field (H) dependence of the thermal conductivity of Bi 2 Sr 2 CaCu 2 O 8+δ (BSCCO) showed a "plateau" in high fields, and they proposed that this plateau region may correspond to a fully-gapped state where there are no QPs to carry heat. This experiment attracted a lot of attention because the fieldinduced phase transition in the superconducting state has a significant implication on the mechanism of the superconductivity; however, the origin of the plateau, as well as the reproducibility of the experimental result, are currently under intense debate (see Ref. [9] and references therein). In this Letter, we report a striking in-plane anisotropy of the QP heat transport, which not only sorts out the "plateau" issue but also suggests an unexpected anisotropy in the superconducting gap symmetry. We measure the thermal conductivity κ of high-quality BSCCO single crystals using a conventional steady-state technique [9,10]. The crystals used here are grown by the floating-zone (FZ) technique [11] and are annealed to be slightly underdoped (zero-resistance T c is 90.5 K); their crystallographic axes are determined by the X-ray Laue analysis within the accuracy of a few degrees. To obtain reliable data on the in-plane anisotropy, we cut a piece of high-quality single-domain crystal into three parts and measure the a and b axis transports separately [ Fig. 1(c)]; both the resistivity (ρ a and ρ b ) and the thermal conductivity (κ a and κ b ) are measured on the same samples. Note that in BSCCO the a and b crystallographic axes are along the nodal directions [ Fig. 1(a)]. The magnetic field is applied perpendicular to the ab plane and the H-dependence of κ is measured with the field-cooled procedure [9] to avoid complications that are associated with the vortex-pinning-related hysteresis [12]. Figure 2 displays the temperature dependences of ρ and κ for the a and b directions; the resistivity shows only a small, if any, in-plane anisotropy, while the thermal conductivity demonstrates a non-trivial anisotropy that cannot be attributed to an error in the geometrical factors; namely, the peak height at ∼70 K is smaller along the a axis and yet κ a becomes larger than κ b below ∼30 K. This anisotropy in κ(T ) suggests that the number of thermally-excited QPs is larger for the nodes along the a axis, because the QP population near the nodes is more directly reflected in κ at lower temperatures [13]. (The anisotropy in κ above T c is most likely due to phonons; it has been estimated [9] that phonons are responsible for roughly 2/3 of the total κ at T c .) We note that the uncertainties in the absolute values in Fig. 2 are less than 10% for both ρ and κ.
The magnetic-field dependence of κ reveals more striking in-plane anisotropy of the heat transport. Figure 3 shows the result of our precise H-dependence measurement (where the relative accuracy is ∼0.1% [9]) for κ a and κ b at 12.5 and 10 K up to 6 T. Interestingly, only κ b shows the plateau-like feature, indicating that there cannot be a true "fully-gapped" state where all the nodes are lifted. (If a "full gap" causes the QPs to disappear, there should be no QP heat transport in any direction.) To further investigate the magnetic-field dependence of κ down to lower temperatures and up to higher magnetic fields, we have measured κ a and κ b in a 3 He refrigerator equipped with a 16 T magnet (with a bit worse signal-to-noise ratio). Figures 4(a) and 4(b) show the H-dependences of κ a and κ b , respectively, down to 0.36 K and up to 16 T; the trend is clear that κ a tends to decrease with H while κ b tends to flatten or increase with H. Note that, since the phonons contribute to the heat transport in parallel to the QPs, the observed thermal conductivity is a sum of the QP term κ QP and the phonon term κ ph . For BSCCO, it has been discussed [8,9,12] that more than 90% of the total κ is due to κ ph around 10 K and this κ ph is independent of H; thus, although the H-dependence of the total κ is rather weak in Figs. 4(a) and 4(b), κ QP is actually changing a lot with H.
The physical origin of this anisotropic behavior can be inferred by plotting κ a and κ b at the same temperature together, taking the absolute values for the vertical axis [Figs. 4(c)-4(e)]. In the simple kinetic theory, the QP thermal conductivity κ QP is proportional to both the number of QPs available for the heat transport, N QP , and their mean free path l QP ; namely, κ QP ∝ N QP l QP . In dwave superconductors, the quantized vortices produced by applied magnetic fields are expected to both induce QPs near the nodes and scatter them [14,15,16]. If there are already a large number of QPs along the a-axis in zero field, the increase in N QP due to the vortices may be inconsequential, while the vortex scattering will cause a noticeable reduction in l QP ; as a result, κ QP along the a-axis would tend to decrease with increasing H. On the other hand, if there are only a small number of QPs along the b-axis in zero field, both N QP and l QP along the b-axis will be noticeably changed due to the vortices and the changes in N QP and l QP will tend to cancel each other, leading to a weaker H-dependence along the baxis. (Actually, one of the probable explanations of the "plateau" in κ(H) is that it comes from an exact cancellation of the two effects [15].]) This phenomenology explains the behaviors of κ a (H) and κ b (H) in Figs. 4(c) and 4(d) (7.5 and 3.0 K) and is consistent with their relative magnitude. At lower temperature [ Fig. 4(e)], it appears that the changes in N QP dominate over the changes in l QP , causing κ to increase with H. (Unfortunately, more quantitative analysis of the data is hindered by the lack of proper understanding of the vortex scattering of the QPs in the cuprates, which remains rather controversial [15,16].) Thus, the novel anisotropies in both the temperature dependence and the magnetic-field dependence of the thermal conductivity consistently tell us that there are more QPs near the nodes along the a-axis; this gives strong evidence that the gap structure notably deviates from the d x 2 −y 2 symmetry. Mixing of a small swave component, which the orthorhombic distortion may cause [17], would not produce the observed anisotropy. One simple possibility to make the gap consistent with the observed anisotropy is to introduce "higher harmonics"; for example, for a gap with an expression ∆(k) = ∆ 0 cos 2φ − ∆ 1 sin 4φ, the slope at the nodes is gentle along the a axis and steeper along the b axis [ Fig. 4(f)] [18]. Of course, some non-analytic modulation of the d x 2 −y 2 symmetry may also be possible.
Since the reproducibility of the plateau in κ(H) has been a matter of debate [9], we comment on the reproducibility issue. We reported previously that the plateau was not observed in the majority of samples [9]; recently, we found that this majority of samples were cut along the a-axis, because our crystals usually grow along the a-axis in the FZ furnace. In fact, the κ(H) data published in Ref. [9] agree with the κ a data reported here. To check whether our salient observation reported here is reproducible, we have measured another b-axis sample (Sample B2) that was taken from a different batch where the crystal grew along the b-axis. We have also measured a sample (Sample M) which was cut along the direction 45 • rotated from the a-axis (Cu-O-Cu bond direction). As shown in Fig. 5, Sample B2 (which is optimally-doped with T c = 95 K) reproduces the behavior of the first baxis sample [ Fig. 4(b)], and Sample M shows the behavior which is essentially an average of κ a (H) and κ b (H). These results give confidence in the reproducibility as well as the intrinsic nature of our observation.
It is useful to note that the angle-resolved photoemission experiments have reported almost completely four- fold-symmetric intrinsic Fermi surface for BSCCO [19], and therefore the novel anisotropy of the gap found here is not likely to be due to a Fermi surface anisotropy [20]. (In fact, the normal-state resistivity shows negligibly small in-plane anisotropy [ Fig. 2(a)], which is consistent with the almost-isotropic Fermi surface.) Therefore, the striking anisotropy in the gap structure suggests that the driving force of the gapping, or the "glue" to bind the electrons into Cooper pairs, is anisotropic itself. From this point of view, it would be difficult to imagine that the antiferromagnetic interaction on the nearly-square CuO 2 planes is wholly responsible for the pairing, though it is often considered to be the most likely candidate. On the other hand, the phonons are expected to be anisotropic in BSCCO because of the pronounced modulation structure [21], and thus the role played by the phonons in pairing is an interesting possibility. Another possibility is a role of the charge inhomogeneity [22], which could be related to the isotropic or nematic phases of the electronic liquid crystal [23,24], though the impact of such inhomogeneous structure on the gapping and on the QP behavior is not understood yet. In any case, the novel anisotropy in the gap structure suggested by the present experiment should help clarifying how the superconducting gap is formed in high-T c superconductors.
In summary, we have measured the thermal conductivity κ of BSCCO along the a and b axes in essentially the same high-quality single crystal down to 0.36 K. It is found that the temperature dependence and the magnetic-field dependence of κ both show peculiar inplane anisotropy, and the observed anisotropy is consistent with the interpretation that there are more QPs near the nodes along the a axis. This result gives evidence that the d-wave gap is uniquely distorted in BSCCO, which suggests that the electron pairing is contributed by the interaction that develops a pronounced anisotropy between the a and b axes. In addition, our measurement clarifies that only κ b shows the "plateau" in its magnetic-