Modulation vector of the Fulde-Ferrell-Larkin-Ovchinnikov state in ${\mathrm{CeCoIn}}_5$ revealed by high-resolution magnetostriction measurements

The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state is an exotic superconducting phase formed by Cooper pairs with finite center-of-mass momentum $\mathit{q}$. On theoretical grounds, the superconducting order parameter in the FFLO state is spatially modulated along the $\mathit{q}$ vector, and the emergence of an associated anisotropy is expected at the phase transition from the Abrikosov state to the FFLO state. Here, we report the results of high-resolution magnetostriction measurements for a single crystal of ${\mathrm{CeCoIn}}_5$ around $B\parallel c$. We find two anomalies in the magnetostriction along the $c$ axis, parallel to the magnetic-field orientation. In sharp contrast, this $B_{\mathrm{K}}$ anomaly disappears in the magnetostriction along the $a$-axis direction, perpendicular to the magnetic-field orientation. To explain this uniaxial expansion, we suggest a possibility that the FFLO transition occurs slightly below the upper critical field, and the FFLO modulation vector parallel to the applied magnetic field gives rise to the anisotropic response.

Figure 1

The $c$-axis magnetostriction coefficient ${\lambda }_c\left(B\right)=(\partial L_c/\partial B)/L_c$ at several temperatures for $B\parallel c$. Each set of data is shifted vertically by ${10}^{-4}{\mathrm{T}}^{-1}$ for clarity. Closed (open) circles represent the data taken in the field-increasing (-decreasing) processes. Inset shows a schematic image of the spatial modulation of the order parameter $\mathrm{\Delta }\left(\mathit{r}\right)$ in the FFLO state along the modulation vector $\mathit{q}$.

Figure 2

Thermal expansion $\mathrm{\Delta }{L}_c/L_c$ measured at 5, 4.89, 4.85, 4.8, 4.7, 4.5, 4.3, and 4 T (from bottom to top) for $B\parallel c$. Closed and open circles represent the data taken in the warming and cooling processes, respectively. Inset shows temperature dependence of the thermal expansion coefficient ${\alpha }_c=(\partial L_c/\partial T)/L_c$ along the $c$ axis at 0 T.

Figure 3

(a) Field-temperature phase diagram of ${\mathrm{CeCoIn}}_5$ for $B\parallel c$. (b) Field dependence of $d{\lambda }_c/dB$ at several temperatures. Both increasing and decreasing field data are shown above 0.2 K (only the former at 0.11 K). Each set of data in (b) is shifted vertically by $5\times {10}^{-3}{\mathrm{T}}^{-2}$ for clarity. The positions of $B_{\mathrm{c}2}$ [squares in (a)] and $B_{\mathrm{K}}$ (triangles) are determined by large and small peaks in $d{\lambda }_c/dB$, respectively, as indicated by solid arrows in (b). The symbols shown in red (blue) are determined from the field-increasing (-decreasing) data. Crosses represent the field above which ${\lambda }_c\left(B\right)$ starts to change markedly toward $B_{\mathrm{K}}$ with increasing $B$, possibly corresponding to a boundary between the Abrikosov state and the FFLO state.

Figure 4

Field angle $\varphi$ dependence of (a) $\mathrm{\Delta }{L}_i/L_i$ at 0.11 K and 5 T, and (b) their field-angle derivatives, ${\lambda }_{\varphi ,i}=(\partial L_i/\partial \varphi )/L_i$ ($i=a$ or $c$). Closed (open) symbols represent the data taken in the $\varphi$-increasing (decreasing) process. The gray solid line in (b) represents the simulation spectrum consisting of two Gaussian functions (dashed and dash-dotted lines). Inset in (a) shows a schematic view of the flux lines and the FFLO modulation. The periodic nodal planes, separated by $L=2\pi /q$, run perpendicular to the flux lines that form a vortex lattice in the $ab$ plane. Inset in (b) shows a wider view of ${\lambda }_{\varphi ,i}\left(\varphi \right)$ in the $\varphi$-increasing (-decreasing) process for $\varphi >0$ ($\varphi <0$).

Figure 5

(a) The wave number $q$ normalized by the coherent length and (b) the normalized entropy $S=S_{\mathrm{s}}\left(T\right)/S_{\mathrm{n}}\left(T_{\mathrm{c}}\right)$ as a function of $h=H/H_{\mathrm{c}2}$, taken from Ref. [36]. Here, $S_{\mathrm{s}}$ ($S_{\mathrm{n}}$) is the entropy in the superconducting (normal) state. (c) $|{\lambda }_c|$ as a function of $B$ at $T=0.06\mathrm{K}$ for $B\parallel c$. (d) $|{\alpha }_c|$ as a function of $T$ at $B=4.73\mathrm{T}$ for $B\parallel c$. (e) $|{\lambda }_{\varphi ,c}|$ as a function of the angle $\varphi$ at $B=4.94\mathrm{T}$ and $T=0.11\mathrm{K}$. The yellow and the pink colored regions represent the Abrikosov state and the FFLO state, respectively; the boundary in $|{\lambda }_c|$ and $|{\lambda }_{\varphi ,c}|$ ($|{\alpha }_c|$) is determined by the onset of the $B_{\mathrm{K}}$ anomaly [inferred from the phase diagram in Fig. 3].