Intertwined Orders in Heavy-Fermion Superconductor ${\mathrm{CeCoIn}}_5$

The appearance of spin-density-wave (SDW) magnetic order in the low-temperature and high-field corner of the superconducting phase diagram of ${\mathrm{CeCoIn}}_5$ is unique among unconventional superconductors. The nature of this magnetic $Q$ phase is a matter of current debate. Here, we present the thermal conductivity of ${\mathrm{CeCoIn}}_5$ in a rotating magnetic field, which reveals the presence of an additional order inside the $Q$ phase that is intimately intertwined with the superconducting $d$-wave and SDW orders. A discontinuous change of the thermal conductivity within the $Q$ phase, when the magnetic field is rotated about antinodes of the superconducting $d$-wave order parameter, demands that the additional order must change abruptly, together with the recently observed switching of the SDW. A combination of interactions, where spin-orbit coupling orients the SDW, which then selects the secondary $p$-wave pair-density-wave component (with an average amplitude of 20% of the primary $d$-wave order parameter), accounts for the observed behavior.

Popular Summary

The ability to tune a material’s properties by external means—for example, electric or magnetic fields, pressure, or subtle changes in composition—is highly desirable in technological applications. Many compounds that exhibit such tunability have been shown to exist at a boundary between states of fundamentally different nature (i.e., order). Such states are often observed to compete with one another; however, sometimes the two states can coexist with each other, and the interaction between the two orders leads to a new state of matter. As a result, the two orders become intertwined. Such phenomena have been proposed to occur in cuprate and iron-based superconductors. Here, we present the case of a material that belongs to yet another class of compounds—the heavy-fermion superconductor ${\mathrm{CeCoIn}}_5$, which exhibits three intertwined orders, two superconducting and one magnetic.

We consider a single crystal of ${\mathrm{CeCoIn}}_5$, a unique material that possesses a so-called $Q$ phase—a novel magnetic state—that arises in large magnetic fields when the sample is superconducting. We measure the thermal conductivity of the sample, which is a powerful probe of a superconducting state. Collecting data at temperatures as low as 0.1 K in a rotating magnetic field, we observe sharp jumps of thermal conductivity in the $Q$ phase, and we show that this state possesses three intertwined orders. We demonstrate that the resulting intertwined state of ${\mathrm{CeCoIn}}_5$ can be controlled by changing the direction of the applied magnetic field.

We have demonstrated a tunable superconducting order for the first time, and we expect that our findings will inspire new research and technological applications.

Figure 1

(a) Phase diagram of ${\mathrm{CeCoIn}}_5$, showing the $Q$ phase, based on specific heat measurements [7]. The red data points are obtained from the present measurements, and the details are explained in Fig. 2. (b,c) Schematic diagrams that illustrate switching of the SDW magnetic domain (${\mathbf{Q}}_{\mathrm{SDW}}$) as the magnetic field $\mathbf{H}$ is rotated about [100]. The heat current $\mathbf{J}$ is in the nodal [110] direction. ${\mathbf{Q}}_{\mathrm{SDW}}$ switches to be more perpendicular to $\mathbf{H}$, while lying along the nodes of the $d$-wave order parameter represented by the green curve. The blue circle represents the normal Fermi surface and the magnetization of the SDW points out of the plane. (d) The thermal conductivity of ${\mathrm{CeCoIn}}_5$ $\kappa$ divided by temperature $T$ in the $Q$ phase as a function of the angle $\theta$ between $\mathbf{H}$ and the heat current $\mathbf{J}\parallel [110]$, at 11 T and 108 mK. The magnetic field is rotated between $-90°$ and $+90°$ within the crystallographic $ab$ plane. At 45° and $-45°$, the antiferromagnetic ordering vector ${\mathbf{Q}}_{\mathrm{SDW}}$ switches between (0.44, 0.44, 0.5) and (0.44, $-0.44$, 0.5), as in (b,c) [27]. When ${\mathbf{Q}}_{\mathrm{SDW}}$ switches from ${\mathbf{Q}}_{\mathrm{SDW}}\perp \mathbf{J}$ to ${\mathbf{Q}}_{\mathrm{SDW}}\parallel \mathbf{J}$, the thermal conductivity increases by approximately 15%. (e) Hysteretic behavior of the thermal conductivity in the switching region around $\theta =-45°$, showing a first-order-like anomaly, for several fields. (f) Hysteretic behavior around $\theta =45°$. The inset shows the width of the hysteresis as a function of magnetic field [from (e)].

Figure 2

(a) Temperature dependence of the thermal conductivity over $T$ ($\kappa /T$) for $\theta =44.7°$ when ${\mathbf{Q}}_{\mathrm{SDW}}\perp \mathbf{J}$ (blue symbols), and for $\theta =45.7°$ when ${\mathbf{Q}}_{\mathrm{SDW}}\parallel \mathbf{J}$ (red symbols) at a fixed field. The step at 0.6 K reflects the superconducting–normal transition. The data with higher thermal gradients (cyan and magenta triangles) show no difference from the other data ($\mathrm{\Delta }T/T\approx 0.06$); i.e., there are no indications of the sliding mode of the SDW (Appendix pp4). Inset: The difference ($\mathrm{\Delta }{\kappa }_{\parallel ,\perp }/T$) between $\kappa /T$ for the two orientations of ${\mathbf{Q}}_{\mathrm{SDW}}$ (green circles, left axis) and the width of the hysteresis $\mathrm{\Delta }\theta$ (from Fig. 6) at two temperatures for ${\mu }_0\mathrm{H}=11\mathrm{T}$ (orange triangles, right axis). The onset temperature of the $Q$ phase at ${\mu }_0\mathrm{H}=11\mathrm{T}$ is depicted as a diamond on the phase diagram in Fig. 1. (b) Magnetic-field dependence of the thermal conductivity over $T$ ($\kappa /T$) at several temperatures. The field directions for the two different ${\mathbf{Q}}_{\mathrm{SDW}}$ orientations are the same as in (a). The inset shows the difference between $\kappa /T$ for the two orientations. The onset of the rise in $\mathrm{\Delta }{\kappa }_{\parallel ,\perp }/T$ is taken as a $Q$-phase transition and is displayed as red circles in Fig. 1.

Figure 3

(a) Magnetic-field dependence of the thermal conductivity over $T$ ($\kappa /T$) in and around the $Q$ phase. The field directions are the same as in Fig. 2. The data for field sweeps down (circles) and up (diamonds) are shown. The difference ($\mathrm{\Delta }{\kappa }_{\parallel ,\perp }/T$) for the two orientations of ${\mathbf{Q}}_{\mathrm{SDW}}$ (green triangles, right axis) starts to grow above 9.9 T and is well described by the $\sqrt{(H-H_c)/H_c}$ fit shown (orange curve). Weak hysteresis within the $Q$ phase is likely due to a vortex-lattice transition for $\mathbf{H}\parallel [100]$ at ${\mu }_0$$\mathrm{H}\approx 11\mathrm{T}$, observed recently with scanning tunneling microscopy [33]. (b) $\kappa /T$ for $\theta =-90°$ when ${\mathbf{Q}}_{\mathrm{SDW}}\parallel \mathbf{J}$ (red symbols), and $\theta =0°$ when ${\mathbf{Q}}_{\mathrm{SDW}}\perp \mathbf{J}$ (blue symbols). The data are very similar to those for the same relative orientation of ${\mathbf{Q}}_{\mathrm{SDW}}$ and $\mathbf{J}$, correspondingly colored, in (a). The scales of the $y$ axes in (a) and (b) are the same but with different offsets. The features for both $\theta =-90°$ and $\theta =0°$ between 8 and 9 tesla are likely due to a vortex-lattice transition for $\mathbf{H}\parallel [110]$ between 7.5 and 8.7 T [34].

Figure 4

Schematics of the $d$-wave (green) and the $p$-wave PDW (orange, described by the vector ${\mathbf{d}}_1$) superconducting order parameters, the ${\mathbf{Q}}_{\mathrm{SDW}}$ vectors and the corresponding gaps (magenta arrow and arcs, respectively), and the magnetic-field directions (cyan arrow). A combination of a SDW [28] and a $p$-wave PDW [14, 27] ${\mathbf{d}}_1$ component is consistent with the thermal conductivity data. (a) When ${\mathbf{Q}}_{\mathrm{SDW}}\perp \mathbf{J}$, the $p$-wave antinodes gap the nodes of the $d$ wave along $\mathbf{J}$ (black arrow), sharply reducing the thermal conductivity. (b) The effect of the SDW gapping the nodes along $\mathbf{J}$ must be smaller (by a factor of approximately 2) than the similar effect of the $p$-wave PDW state in (a).

Figure 5

The calculated thermal conductivity as a function of the relative amplitude $a$ of the $p$-wave component for $|{\mathrm{\Delta }}_d|+ia|{\mathrm{\Delta }}_p|$ pairing symmetry without normalization for two orientations of the $p$-wave component ${\mathbf{d}}_1$ with respect to the heat current $\mathbf{J}$ depicted in Figs. 4 and 4 of the main text. The nodes of the $p$-wave component are either perpendicular to $\mathbf{J}$ [Fig. 4] or parallel to it [Fig. 4]. The electron mean-free path $l=10\xi$, where $\xi$ is the superconducting coherence length, $T=0.05T_c$, and $H=0.3H_{c2}$, where $H_{c2}$ is the orbital upper critical field. The reduction of $\kappa$ is much stronger when the $p$-wave antinode is along the heat current ($p$-wave $\text{nodes}\perp \mathbf{J}$) because the $p$-wave antinode gaps the $d$-wave nodal quasiparticles with momenta along the heat current $\mathbf{J}$. In contrast, when the $p$-wave antinode is perpendicular to $\mathbf{J}$ ($p$-wave $\text{nodes}\parallel \mathbf{J}$), the $p$-wave antinode only gaps quasiparticles with momenta perpendicular to the heat current, resulting in a much smaller effect. The thermal transport in a $d$-wave superconductor is therefore dominated by the quasiparticles in the nodes that are along the heat current.

Figure 6

The hysteresis of the hypersensitive switching via thermal conductivity around $\theta =-45°$ at 11 T and two temperatures, 106 and 199 mK. The data for 106 mK are the same as in Fig. 1 of the main text. The widths of the hysteresis at two different temperatures are plotted in the inset of Fig. 2.

Figure 7

Possible configuration of a FFLO state in ${\mathrm{CeCoIn}}_5$. Schematic of the $d$-wave and the FFLO order parameters, the ${\mathbf{Q}}_{\mathrm{SDW}}$ vector, and the magnetic-field directions that are compatible with both hypersensitivity and thermal conductivity within a modified version of the FFLO scenario [29]. This modification would require that ${\mathit{q}}_{\mathrm{FFLO}}$ be forced to lie along the nodes of the $d$ wave, instead of always being parallel to the applied magnetic field. Dashed lines indicate the nodes of the FFLO state with normal quasiparticles. (a) FFLO nodal planes increase scattering and reduce thermal conductivity when they are perpendicular to $\mathbf{J}$. (b) The nodal planes contribute to thermal transport along $\mathbf{J}$ when they are parallel to it.