Quasiparticle Entropy in the High-Field Superconducting Phase of ${\mathrm{CeCoIn}}_5$

The heavy-fermion superconductor ${\mathrm{CeCoIn}}_5$ displays an additional transition within its superconducting (SC) state, whose nature is characterized by high-precision studies of the isothermal field dependence of the entropy, derived from combined specific heat and magnetocaloric effect measurements at temperatures $T\ge 100\mathrm{mK}$ and fields $H\le 12\mathrm{T}$ aligned along different directions. For any of these conditions, we do not observe an additional entropy contribution upon tuning at constant temperature by magnetic field from the homogeneous SC into the presumed Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) SC state. By contrast, for $H\parallel [100]$ a reduction of entropy was found that quantitatively agrees with the expectation for spin-density-wave order without FFLO superconductivity. Our data exclude the formation of a FFLO state in $\mathrm{CeCoI}{\mathrm{n}}_5$ for out-of-plane field directions, where no spin-density-wave order exists.

Figure 1

(a) Magnetic field dependence of electronic specific heat divided by temperature (black solid circles, left axis) and magnetic Grüneisen ratio (red open circles, right axis) of ${\mathrm{CeCoIn}}_5$ at 0.2 K for $H\parallel [100]$. (b) Calculated field derivative of the entropy (red open circles, right axis) and integrated entropy increment (black solid circles, left axis). The inset displays field dependence of magnetic Grüneisen ratio at various different temperatures. Arrows indicate the transition to the high-field SC phase, $H_{\mathrm{HF}}$.

Figure 2

(a) Electronic specific heat divided by temperature of ${\mathrm{CeCoIn}}_5$ at 0.2 K for $H\parallel [100]$ (solid circles) and 18° tilted from [100] toward [001] (open circles) versus magnetic field normalized by upper critical field $H_{c2}=11.7\mathrm{T}$ ($H\parallel [100]$) and 9.0 T ($H$ tilted by 18° toward [001]), respectively. The inset shows the respective magnetic Grüneisen ratio data. (b) Calculated field derivative of the entropy. The inset shows the respective entropy increments for $H\parallel [100]$ (solid circles) and 18° (open circles). Solid arrows indicate the high-field transition for $H\parallel [100]$. The two-sided open arrow in the inset indicates the entropy difference of $8\mathrm{mJ}/\mathrm{mol}\mathrm{K}$.

Figure 3

(a) Electronic specific heat divided by temperature (black solid circles, left axis) and magnetic Grüneisen ratio (red open circles, right axis) of ${\mathrm{CeCoIn}}_5$ as a function of a field for $H\parallel [001]$ at 0.2 K. The inset shows the superconducting phase diagram for $H\parallel [001]$. Black circles and red squares mark the positions of upper critical field and “kink anomaly” (cf., black arrows in main panels), respectively. (b) Calculated field derivative of the entropy $(dS/dH{)}_T$ (red open circles, right axis) and integrated entropy increment (black solid circles, left axis). The inset shows $(dS/dH{)}_T$ for the field applied along [001] and different angles tilted away from [100] toward [001]. Dashed line, see the text.