Resonances in the Field-Angle-Resolved Thermal Conductivity of ${\mathrm{CeCoIn}}_5$

The thermal conductivity measurement in a rotating magnetic field is a powerful probe of the structure of the superconducting energy gap. We present high-precision measurements of the low-temperature thermal conductivity in the unconventional heavy-fermion superconductor ${\mathrm{CeCoIn}}_5$, with the heat current $\mathit{J}$ along the nodal [110] direction of its $d_{x^2-y^2}$ order parameter and the magnetic field up to 7 T rotating in the $ab$ plane. In contrast to the smooth oscillations found previously for $\mathit{J}\parallel [100]$, we observe a sharp resonancelike peak in the thermal conductivity when the magnetic field is also in the [110] direction, parallel to the heat current. We explain this peak qualitatively via a model of the heat transport in a $d$-wave superconductor. In addition, we observe two smaller but also very sharp peaks in the thermal conductivity for the field directions at angles $\mathrm{\Theta }\approx \pm 33°$ with respect to $\mathit{J}$. The origin of the observed resonances at $\mathrm{\Theta }\approx \pm 33°$ at present defies theoretical explanation. The challenge of uncovering their source will dictate exploring theoretically more complex models, which might include, e.g., fine details of the Fermi surface, Andreev bound vortex core states, a secondary superconducting order parameter, and the existence of gaps in spin and charge excitations.

Figure 1

Schematic diagram of the thermal conductivity measurement on ${\mathrm{CeCoIn}}_5$. The heat current $\mathit{J}$ (magenta arrow) was applied along a node of the $d_{x^2-y^2}$ order parameter (green curve). The angle $\mathrm{\Theta }$ is between the magnetic field $\mathbf{H}$ (cyan arrow) and the direction of the heat current. The blue circle represents the normal Fermi surface.

Figure 2

Field-angle dependent thermal conductivity at different temperatures and with a fixed 1-T magnetic field. The solid lines represent fits to the equation $f(\theta )=a_0+a_2\cos (2\theta )+a_4\cos (4\theta )$ and the coefficients are plotted in the inset.

Figure 3

Field-angle dependent thermal conductivity at three different temperatures with a fixed 3-T magnetic field.

Figure 4

Field-angle dependent thermal conductivity at 0.11 K for several magnetic fields between 1 and 7 T. The gray curve is a calculation based on the model of Ref. [14], see the Supplemental Material [22].

Figure 5

Expanded view of the anomalies around (a) $-33$° and (b) 33°. (c) The intensities of the peaks at $\mathrm{\Theta }\approx \pm 33°$ as a function of magnetic field. The detailed procedures of extracting the peak intensities are described in the Supplemental Material [22].