Superconducting gap function in antiferromagnetic heavy-fermion $\mathrm{U}{\mathrm{Pd}}_2{\mathrm{Al}}_3$ probed by angle-resolved magnetothermal transport measurements

The superconducting gap structure of heavy fermion $\mathrm{U}{\mathrm{Pd}}_2{\mathrm{Al}}_3$, in which unconventional superconductivity coexists with antiferromagnetic (AF) order with atomic size local moments, was investigated by the thermal conductivity measurements in a magnetic field $\mathit{H}$ rotating in various directions relative to the crystal axes. The thermal conductivity displays distinct twofold oscillation when $\mathit{H}$ is rotated in the plane orthogonal to the basal ab plane, while no oscillation was observed when $\mathit{H}$ is rotated within the basal plane. These results provide strong evidence that the gap function $\mathrm{\Delta }\left(\mathit{k}\right)$ has a single line node orthogonal to the $c$ axis located at the AF Brillouin zone boundary, while $\mathrm{\Delta }\left(\mathit{k}\right)$ is isotropic within the basal plane. This gap structure indicates that the pairing interaction in neighboring planes strongly dominates over the interaction in the same plane. The determined nodal structure is compatible with the resonance peak in the dynamical susceptibility observed in neutron inelastic scattering experiments. Based on these results, we conclude that the superconducting pairing function of $\mathrm{U}{\mathrm{Pd}}_2{\mathrm{Al}}_3$ is most likely to be $d$-wave with a form $\mathrm{\Delta }\left(\mathit{k}\right)={\mathrm{\Delta }}_0\cos \left(k_{z}c\right)$.

Figure 1

Temperature dependence of the thermal conductivity in zero field in the $b$ axis ${\kappa }_{yy}(\mathit{q}\parallel b)$ and $c$ axis ${\kappa }_{zz}(\mathit{q}\parallel c)$. Inset shows the structure of the hexagonal basal plane of $\mathrm{U}{\mathrm{Pd}}_2{\mathrm{Al}}_3$ with the alignment of the $a$ axis (100) and $b$ axis (−1,2,0), as used in the text.

Figure 2

Magnetic field dependence of the $b$-axis thermal conductivity ${\kappa }_{yy}$ for $\mathit{H}\parallel a$ and $\mathit{H}\parallel c$. ${\kappa }_{yy}$ is normalized by the normal state value just above the upper critical field ${\kappa }_{yy}^n$.

Figure 3

(a) Schematic figure of the gap structure with four line nodes perpendicular to the basal plane (vertical node). (b) Schematic diagram showing the regions on the Fermi surface that experience the Doppler-shift in $\mathit{H}$ within the basal plane. We have assumed $d_{xy}$ symmetry. $\varphi =(\mathit{H},a)$ is the azimuthal angle measured from the $a$ axis. With $\mathit{H}$ applied along the antinodal directions, all four nodes contribute to the DOS, while for $\mathit{H}$ applied parallel to the node directions, the Doppler shift vanishes at two of the nodes. (c) Fourfold oscillation of the DOS for $\mathit{H}$ rotating in the basal plane. The DOS shows a maximum (minimum) when $\mathit{H}$ is applied in the anitinodal (nodal) direction.

Figure 4

(a) Schematic figure of the gap structure with line nodes parallel to the basal plane (horizontal node). Line nodes are located at the bottleneck (type I) and at the AF zone boundary (type II). Two line nodes are located at the bottleneck and the AF zone boundary (type III) and at positions shifted off the bottleneck (type IV). (b) Oscillations of the DOS for $\mathit{H}$ rotating in the ac plane for various gap functions. Twofold oscillation with the same sign are expected for type I, II, and III. On the other hand, for type IV, an oscillation with a double minimum is expected.

Figure 5

Angular variation of the $c$-axis thermal conductivity ${\kappa }_{zz}(\mathit{H},\varphi )$ normalized by the normal state value, ${\kappa }_{zz}^n$, at several fields at 0.4 K. $\mathit{H}$ was rotated within the basal plane (see inset). A distinct sixfold oscillation is observed above 0.5 T, while oscillation is absent at 0.5 and 0.3 T. At $H=3.5\mathrm{T}(>H_{c2})$, the system is in the normal state.

Figure 6

Field dependence of the amplitude of the sixfold oscillation $C_{zz}^{6\varphi }$ normalized by the normal state value, ${\kappa }_{zz}^n$. $H_{c2}^a=3.2\mathrm{T}$ and $H_{c2}^b=3.12\mathrm{T}$ are the upper critical fields in $\mathit{H}$ parallel to the $a$ and $b$ axis, respectively. The inset shows the magnetic domain structures in parallel fields. The hexagon shows the basal plane. Populated magnetic domains with nonparallel spins are indicated by filled arrows. The open arrows indicate the antiparallel spins in the neighboring plane. The transition (i)–(ii) occurs at about 0.6 T. In the (i) phase, the ordered moments point to the $a$ axis, forming domain structures with sixfold degeneracy. In the (ii) phase, the domain structure changes to sixfold symmetry with respect to the $\mathit{H}$ rotation in the basal plane.

Figure 7

Angular variation of the $b$-axis thermal conductivity ${\kappa }_{yy}(\mathit{H},\theta )$ normalized by the normal state value ${\kappa }_{yy}^n$ at several fields at 0.4 K. $\mathit{H}$ was rotated within the ac-plane perpendicular to the basal plane (see inset).

Figure 8

(a) $H$ dependence of the amplitude of the twofold symmetry $C_{yy}^{2\theta }$ normalized by the normal state thermal conductivity ${\kappa }_{yy}^n$ at 0.4 K. The sign of $C_{yy}^{2\theta }$ is negative in the (I) and (III) regions, and is positive in the (II) region. (b) Field dependence of the $b$-axis thermal conductivity ${\kappa }_{yy}$ for $\mathit{H}\parallel c$ at 0.36 K.

Figure 9

Angular variation of the $b$-axis thermal conductivity ${\kappa }_{yy}^{2\theta }∕{\kappa }_{yy}^n$ with rotating $\mathit{H}$ within the ac plane and of the $c$-axis thermal conductivity ${\kappa }_{zz}^{6\varphi }∕{\kappa }_{zz}^n$ with rotating $\mathit{H}$ within the basal ab plane at $T=0.4\mathrm{K}$ and at $H=0.5\mathrm{T}$. The amplitude of the sixfold oscillation in ${\kappa }_{zz}^{6\varphi }∕{\kappa }_{zz}^n$ is less than 0.2% if it exists. Inset: Schematic figure of the gap function of $\mathrm{UP}{\mathrm{d}}_2{\mathrm{Al}}_3$ determined by angle-resolved magnetothermal transport measurements. The thick solid lines indicate horizontal nodes located at the AF zone boundaries.