The hexagonal heavy-fermion compound UPd₂Al₃ exhibits superconductivity below T_c = 2 K, which coexists with an antiferromagnetic ordering that sets in at T_N = 14.5 K [1]. Various experiments including inelastic neutron scattering [2] and thermal conductivity [3] measurements have inferred that the superconducting gap function is of nodal A_1g type, Δ(k) = Δ₀cos(k_zc), having horizontal line nodes. Up to present, however, no thermodynamic evidence for the horizontal line node has been obtained.

(a) Field-angle variation of C(H,θ)/T of UPd₂Al₃ with H rotated in the ac plane measured at 0.2 K. (b) Calculated results of the field-angle dependence of ZEDOS in magnetic fields normalized by H_c2 for a linear horizontal line node. The inset shows the definition of angles for the hexagonal structure.

In order to examine the pairing symmetry of UPd₂Al₃ in the light of quasiparticle density of states, we performed angle-resolved heat capacity (C) measurements on a high quality single crystal of UPd₂Al₃ in rotating magnetic fields (H) [4]. When H is rotated in the basal plane, no angular variation of C(H,ϕ) that could be ascribed to vertical line nodes is observed. To probe the horizontal line node, we examined the polar-angle (θ) dependence C(H,θ) and the results are shown in Fig. 1(a). Below 0.5 T, a twofold oscillation is clearly observed with a maximum at θ = 0˚ (H||[0001]) and a minimum at θ = 90˚ (H||[1120]). Above 2.5 T, the twofold angular oscillation is reversed and the maximum appears along [1120], reflecting the underlying nodal structure as well as the anisotropy of the upper critical field H_c2. Very interestingly, in the field range from 1 to 2 T, a new feature appears in C(H,θ) in an intermediate angle region; at 1 T, a shoulder or a hump appears at θ ~ 30˚, and moves to the higher angle side θ ~ 45˚-60˚ with increasing H to 2 T. Above 2.5 T, it finally merges into the maximum at θ = 90˚ arising from the H_c2 anisotropy. Remarkably, the maximum of C(H,θ)/T at 1.5 T occurs near 45˚.

The observed field evolution of C(H,θ), in particular the shoulder/hump structure in the intermediate angular region, provides strong evidence that a horizontal line node exists on the Fermi surface. We performed microscopic calculations of the zero-energy density of states (ZEDOS) by means of the quasiclassical Eilenberger theory within the Kramer-Pesch approximation [4]. Figure 1(b) shows the calculated angular dependence of ZEDOS, assuming a spherical Fermi surface and a model gap function ∆(k) = ∆₀k_z that has a horizontal line node at the equator. The results clearly demonstrate that the anisotropy inversion occurs in the ZEDOS as H increases. What is more important is that the calculated results successfully reproduce the shoulder/hump anomaly around 30-60˚ in C(H,θ) at the intermediate fields. The polar angle dependence of C(H,θ), therefore, possesses sufficiently high resolution to detect the horizontal line node.

References

• [1] C. Geibel et al., Z. Phys. B Condens. Matt. 84, 1 (1991).
• [2] N. Bernhoeft et al., Phys. Rev. Lett. 81, 4244 (1998).
• [3] T. Watanabe et al., Phys. Rev. B 70, 184502 (2004).
• [4] Y. Shimizu et al., Phys. Rev. Lett. 117, 037001 (2016).

Authors

• Y. Shimizu, S. Kittaka, T. Sakakibara, Y. Tsutsumi^a, T. Nomoto^b, H. Ikeda^c, K. Machida^c, Y. Homma^d, and D. Aoki^d
• ^aUniversity of Tokyo
• ^bKyoto University
• ^cRitsumeikan University
• ^dTohoku University