Subphases in the superconducting state of ${\mathrm{CeIrIn}}_5$ revealed by low-temperature $c$-axis heat transport

Low-temperature (down to $\sim 50$ mK) thermal conductivity measurements with the heat flow direction along the interplane tetragonal $c$ axis, ${\kappa }_c$, were used to study the superconducting state of heavy fermion ${\mathrm{CeIrIn}}_5$. Measurements were performed in the magnetic fields both parallel to the heat flow direction $H\parallel c$, and transverse to it $H\parallel a$. Interplane heat conductivity in $H\parallel c$ configuration shows negligible initial increase with magnetic field and a rapid rise on approaching $H_{c2}$ from below, similar to the expectations for the superconducting gap without line nodes. This observation is in stark contrast to monotonic increase found in the previous in-plane heat transport measurements. In the configuration with the magnetic field breaking the tetragonal symmetry of the lattice $H\parallel a$, ${\kappa }_c$ reveals nonmonotonic evolution with temperature and magnetic field suggesting subphase boundary in the superconducting state. The characteristic temperature $T_{\mathrm{kink}}\sim$ 0.07 K of the subboundary is well within the domain of bulk superconductivity $T_c\sim$ 0.4 K and $H_{c2}\sim$ 1.0 T. These results are consistent with a superconducting gap with an equatorial line node and polar point nodes, a gap symmetry of the ${\mathrm{D}}_{4h}$ point group, for which magnetic field along the tetragonal plane breaks the degeneracy of the multicomponent order parameter and induces a phase transition with nodal topology change.

Figure 1

Temperature-dependent interplane thermal conductivity, plotted as ${\kappa }_c/T$ vs $T$, over the whole range up to zero-resistance $T_c=$ 0.8 K (a) and over range of bulk superconductivity $T_c=$ 0.4 K (b) in magnetic fields parallel to $c$ axis along the heat current. Blue line in panel (a) shows phonon contribution ${\kappa }_{\mathrm{ph}}/T$. Bottom to top: 0 T (black-solid circles), 0.05 T (green-solid circles), 0.07 T (dark-yellow open diamonds), 0.1 T (blue-open diamonds), 0.2 T (red-solid diamonds), 0.3 T (magenta-open hexagons), 0.4 T (magenta-solid squares), 0.44 T (olive-open up-triangles), and 0.5 T (normal state, black-open circles). In (c) the data are plotted normalized by the normal state 0.5 T curve, $\kappa /{\kappa }_N$, clearly showing convergence of curves above bulk $T_c=$ 0.4 K. (d) Field-dependent normalized thermal conductivity ${\kappa }_s/{\kappa }_N$ taken at 0.1 K (olive-solid up-triangles), at base temperature of our experiment 56 mK (black-open circles) and in the $T^2$ extrapolation $T\to$ 0 (blue-solid circles).

Figure 2

Temperature-dependent interplane resistivity, measured in zero magnetic field (black-solid line) and in magnetic field 4 T parallel to $c$ axis (above resistive $H_{c2\rho }$, olive-solid line). For comparison we plot thermal analog of the electrical resistivity, $w\equiv L_{0}T/\kappa$ in zero field (black-solid circles) and in the magnetic fields above bulk $H_{c2}=$ 0.5 T (red-open circles) and resistive $H_{c2\rho }=$ 4 T (olive-open up-triangles). The $w$ and $\rho$ curves for $H=$ 4 T converge at $T\to$ 0 as expected for the Wiedemann-Franz law. The $w\left(T\right)$ data at 0.5 T show the same temperature dependence as 4 T curve shifted to the lower values due to the magnetoresistance. Inset shows $\rho \left(T\right)$ curve over the whole temperature range up to room temperature.

Figure 3

Temperature-dependent interplane thermal conductivity, plotted as ${\kappa }_c/T$ vs $T$, over the whole temperature range up to zero-resistance $T_c=$ 0.8 K (a) and zoom at the lowest temperatures below 0.2 K (b) in magnetic fields parallel to the $a$ axis in the tetragonal plane transverse to the heat current. Bottom to top: 0 T (black-solid circles), 0.05 T (green-solid circles), 0.1 T (blue-open diamonds), 0.2 T (red-solid diamonds), 0.3 T (magenta-open hexagons), 0.6 T (olive-open up-triangles), 0.8 T (purple-solid down-triangles), and 1 T (normal state, open-black circles). Dashes show $T^2$ fit of the last two data points for all field values, grey-solid line shows fitting range dependence of the $T\to 0$ extrapolation using last four data points for zero-field curve, illustrating the error bar evaluation. Arrows in panel (b) show onset of the rapid decrease of thermal conductivity below the characteristic temperature $T_{\mathrm{kink}}$. In the left-bottom panel (c) the data are plotted normalized by the normal state curve at 1 T, $\kappa /{\kappa }_N$. Bottom-right panel (d) shows field-dependent normalized thermal conductivity ${\kappa }_s/{\kappa }_N$ taken at 0.1 K (olive-solid up-triangles), at 68 mK (light-cyan stars), at base temperature of our experiment 57 mK in $T-$ sweep mode (black-open circles), in $H-$ sweep mode at 52 mK (brown-solid down-triangles), and in the $T^2$ extrapolation to $T\to$ 0 (red-solid circles).

Figure 4

Residual linear terms determined with $T^2$ extrapolation of the data (solid symbols) ${\kappa }_0/T$ in the $c$-axis thermal conductivity of ${\mathrm{CeIrIn}}_5$, plotted on scales normalized to the normal state $\kappa /{\kappa }_N$ vs $H/H_{c2}$. Red circles are for $H\parallel a$, blue squares for $H\parallel c$. Green-solid line shows the field dependence for the in-plane heat transport in $H\parallel c$ configuration [52]. For reference we plot the actual data at the base temperatures of our experiment (open symbols) and the standard dependencies (dashes) observed in the $s$-wave superconductors, as in the clean Nb and the dirty InBi shown here (reproduced from Ref. [52]) and for the $d$-wave (nodal) superconductor Tl-2201 [53].

Figure 5

Comparison of the $c$-axis thermal conductivity ${\kappa }_c/T$ of ${\mathrm{CeIrIn}}_5$ taken in equivalent normalized magnetic fields $H/H_{c2}\sim$ 0.2 for which the difference between the two configurations is maximum, Fig. 4, in $H\parallel c=$ 0.1 T and $H\parallel a=$ 0.2 T. The difference in the behavior for the two field configurations is obvious.

Figure 6

Comparison of the field-dependent interplane thermal conductivity ${\kappa }_c$ with the heat capacity measurements of Ref. [35] both taken in the configuration with magnetic field parallel to the plane $H\parallel a$. For both measurements we normalize the data by their normal state values and use a dimensionless field scale $H/H_{c2}$. Relatively high base temperature in the heat capacity measurements, 80 mK or 0.2 $T_c$, leads to the value of the residual heat capacity in zero field amounting to the half of the normal state value. For reference we show thermal conductivity taken at the closest temperatures 0.1 K (solid triangles) and 57 mK. Note sharp feature in the field-dependent heat capacity at small fields, denoted with arrow, and plotted in the phase diagram in Fig. 7 below.

Figure 7

The $H-T$ phase diagram of the superconducting state of ${\mathrm{CeIrIn}}_5$ as revealed by the interplane heat transport in the magnetic fields parallel to the tetragonal conducting plane $H\parallel a$. The $T_c\left(H\right)$ points (red-open circles and solid diamonds) were determined using steep kinks at the transition temperature in $\kappa /T$, Fig. 3, or in ${\kappa }_s/{\kappa }_N$, Fig. 3, these data are in reasonable agreement with the heat capacity data (black-open diamonds) revealing much clearer anomalies at $T_c$ [35]. The temperature of the kink feature $T_{\mathrm{kink}}$ (open up-triangles) was determined at onset of the decrease in ${\kappa }_c/T$ vs $T$ [arrows in Fig. 3]. This feature is linked with the rapid rise of $T\to 0$ extrapolation in the field dependence. The low field-low temperature phase, below 0.07 K and 0.05 T, has strong $ac$ plane gap anisotropy [25, 34], big blue point corresponds to an onset of the anisotropy increase on cooling. This suggests its relation to the hybrid $E_g(1,i)$ symmetry [25]. The orange-solid diamond point is from heat capacity data Ref. [35], see Fig. 6. The high field-low temperature phase has a notably increased residual conductivity, suggestive of transformation of $E_g(1,i)$ state to a lower symmetry $E_g(1,0)$ or $E_g(0,1)$ state with vertical line nodes. Whether the high field boundary of the phase (dashed lines) is going all the way to $H_{c2}$, or terminates before it, is not defined in our experiment. The high-temperature phase with fourfold symmetry in the thermal conductivity and the heat capacity measurements might have $d$-wave symmetry [26, 35]. Note, the phase diagram is consistent with the theoretical accidental degeneracy models [63]. Although, according to the theoretical 2D representation models [60] $E_g(1,0)$ or $E_g(1,1)$ symmetries are also possible.