Multiple phases with intertwined magnetic and superconducting orders in Nd-doped ${\mathrm{CeCoIn}}_5$

The thermal conductivity of heavy-fermion superconductor ${\mathrm{Ce}}_{0.95}{\mathrm{Nd}}_{0.05}{\mathrm{CoIn}}_5$ was measured with a magnetic field rotating in the tetragonal $a\text{−}b$ plane. The thermal conductivity exhibits a step within the high-field low-temperature (HFLT) phase, which exists above 8 Tesla within the superconducting state, when field is rotated through the antinodal [100] direction of the superconducting $d$-wave order parameter. This anomaly indicates the presence of a third order parameter within the HFLT phase, similar to that of pure ${\mathrm{CeCoIn}}_5$. Therefore, the HFLT state with triply intertwined orders, i.e., superconducting $d$-wave, magnetic spin-density wave (SDW), and a putative superconducting $p$-wave pair-density wave (PDW), is robust against 5% Nd doping. The second magnetic phase within the superconducting state below 8 Tesla, a low-field SDW, displays a hysteresis in thermal conductivity as a function of field magnitude, which is also revealing the existence of a third order consistent with a $p$-wave PDW. Given different origins and properties of the HFLT and low-field SDW states, the presence of three intertwined orders in both phases is remarkable and of far-reaching consequences.

Figure 1

(a) The thermal conductivity $\kappa$ of 5% Nd-doped ${\mathrm{CeCoIn}}_5$ divided by temperature $T$ as a function of temperature in several magnetic fields. Arrows depict the onsets of the superconducting transition. Inset: Schematic phase diagram of 5% Nd-doped ${\mathrm{CeCoIn}}_5$ [6]. Superconducting–normal transition points are acquired in the present study. Red diamonds are from field sweeps and blue dots are from temperature sweeps. (b) $\kappa /T$ as a function of field for several temperatures. All scans are performed with decreasing magnetic field. The evolution of $\kappa /T$ near $H_{c2}$ demonstrates the first-order nature of the superconducting transition at low temperature. Inset: The change in thermal conductivity $\mathrm{\Delta }\kappa$ from just below to just above $H_{c2}$ divided by the thermal conductivity in the normal state ${\kappa }_N$ just above $H_{c2}$. The direction of the magnetic field for both (a) and (b) is $5^{\circ }$ away from the [100] direction in the $a\text{−}b$ plane ($\theta ={40}^{\circ }$).

Figure 2

The dependence of the thermal conductivity divided by temperature ($\kappa /T$) on the angle $\theta$ between the applied field $\mathbf{H}$ and the heat current $\mathit{J}$ that is along [110]. (a) $\kappa /T$ at 0.15 K for several values of field, in both phases with SDW order. Additional data for wider angles are presented in Fig. 5 (Appendix pp2). (b) Thermal conductivity jumps when a magnetic field of 10 T is rotated though the [100] ($\theta ={45}^{\circ }$) direction. Inset: The difference between the thermal conductivities of field-down scans at $\theta ={40}^{\circ }$ and $\theta ={50}^{\circ }$.

Figure 3

The thermal conductivity divided by temperature ($\kappa /T$) at 0.15 K measured while sweeping field from 0 to 12 T and then back to 0 T for four different field directions. Hysteresis is observed around 4 T in all but one ($\theta ={40}^{\circ }$) directions. No hysteresis is observed above 6 T within the experimental uncertainty. Insets: Diagrams of the experimental setup and the intertwined orders. Blue circle: schematic Fermi surface; Green: $d_{x^2-y^2}$ superconducting order parameter; Pink arrow: SDW ordering wave vector $\mathbf{Q}$; Orange: $p$-wave PDW superconducting order parameter; Cyan arrow: applied magnetic field $\mathbf{H}$; Black arrow: heat current $\mathit{J}$; $\theta$ is the angle between the heat current and the magnetic field. Both $\mathbf{Q}$ and the $p$-wave order flip ${90}^{\circ }$ when the field is switched between two perpendicular nodal directions in (a) and (d).

Figure 4

The field-angle dependence of $\kappa /T$ near $\mathbf{H}\parallel \left[100\right]$ with 1-T (a) and 6-T (b) magnetic field.

Figure 5

The field-angle dependence of $\kappa /T$ between $\theta =-{90}^{\circ }$ and $\theta ={90}^{\circ }$.

Figure 6

The magnetic-field dependence of $\kappa /T$. The inset shows an expanded view of the data near zero field, which are collected with increasing field from zero-field-cooled states. The data above 1 T are the same as those in Fig. 1. The direction of the magnetic field is $5^{\circ }$ away from the [100] direction in the $a\text{−}b$ plane.