High-field state of the flux-line lattice in the unconventional superconductor ${\mathrm{CeCoIn}}_5$

Ultrasound velocity measurements of the unconventional superconductor ${\mathrm{CeCoIn}}_5$ with extremely large Pauli paramagnetic susceptibility reveal an unusual structural transformation of the flux-line lattice (FLL) in the vicinity of the upper critical field. The transition field coincides with that at which heat capacity measurements reveal a second-order phase transition. The lowering of the sound velocity at the transition is consistent with vortex segmentation and a crossover to quasi-two-dimensional FLL pinning. These results provide strong evidence that the high-field state is the Fulde-Ferrel-Larkin-Ovchinikov phase, in which the order parameter is spatially modulated and has planar nodes aligned perpendicular to the vortices.

Figure 1

The transverse sound velocity $v_t^0$ as a function of temperature in two different configurations. (a) The Lorentz mode $\mathbf{H}\parallel \mathbf{k}\parallel \left[100\right]$, $\mathbf{u}\parallel \left[010\right]$ and (b) the non-Lorentz mode, $\mathbf{u}\parallel \mathbf{H}\parallel \left[100\right]$, $\mathbf{k}\parallel \left[010\right]$. The configuration (a) corresponds to a flux line bending mode. The inset to (b) shows $v_t^0$ in zero field and in the normal state above $H_{c2}^{\Vert }$, including the high-field data in the non-Lorentz mode with $H<H_{c2}^{\Vert }$ ($9.5\mathrm{T}$, $10.8\mathrm{T}$, and $11.6\mathrm{T}$ from right to left).

Figure 2

The relative shift of the transverse sound velocity $\Delta v_t^0∕v_t^0$ as function of $\mathrm{H}$ at $100\mathrm{mK}$ for the Lorentz $\left(\mathbf{H}\perp \mathbf{u}\right)$ and the non-Lorentz modes $(\mathbf{H}\parallel \mathbf{u})$. The crystal stiffening at $T_c$ results in a negative apparent $\Delta v_t^0∕v_t^0$ for $H>H_{c2}$. The inset depicts the difference between Lorentz and non-Lorentz modes, $\Delta v_t=v_t^0(\mathbf{u}\perp \mathbf{H})-v_t^0\left(\mathbf{u}\parallel \mathbf{H}\right)$, which can be attributed to the FLL. The dashed line is a fit to Eq. (1).

Figure 3

The relative shift of the transverse sound velocity arising from the contribution of the FLL, $\Delta v_t∕v_t$, as a function of temperature, below the upper critical field $H_{c2}^{\Vert }$. $\Delta v_t∕v_t$ is obtained as the difference between the non-Lorentz mode and the Lorentz mode sound velocities. For clarity, the zero levels of the different curves are vertically shifted. At $T^{*}$, marked by arrows, $\Delta v_t∕v_t$ exhibits a distinct cusp. The dashed lines are $\Delta v_t∕v_t$ extrapolated from above $T^{*}$. Dotted arrows indicate $T_c\left(H\right)$.

Figure 4

(Color) Experimental $H-T$ phase diagram for ${\mathrm{CeCoIn}}_5$ below $0.7\mathrm{K}$ for $(\mathbf{H}\parallel \left[100\right])$. The transition temperature $T^{*}$, indicated by red squares, was obtained from ultrasonic measurements as indicated by the arrows in Fig. 3. Blue circles and solid line depict the upper critical field $H_{c2}$ determined by the ultrasonic experiments. In this temperature regime, the transition at $H_{c2}^{\Vert }$ is of first order. The dash-dotted line is the second-order transition line determined by the heat capacity reported in Ref. 14. The region shown by green is in the FFLO state. The schematic figures are sketches of a flux line above and below the transition.