Superconducting and ferromagnetic phase diagram of UCoGe probed by thermal expansion

We report thermal expansion measurements on a single crystal of the superconducting ferromagnet UCoGe for magnetic fields applied along the main orthorhombic axes. The thermal expansion cell was mounted on a piezoelectric rotator in order to fine-tune the magnetic field angle. The superconducting and magnetic phase diagram has been determined. With our bulk technique we confirm the $S$ shape of the upper-critical field, $B_{c2}$, for $B\parallel b$ and reinforcement of superconductivity above 6 T. At the same time the Curie point shifts towards lower temperatures on increasing the field along the $b$ axis. This lends further support to theoretical proposals of spin-fluctuation-mediated reinforcement of superconductivity for $B\parallel b$.

Figure 1

Temperature variation of the coefficient of linear thermal expansion of UCoGe along the orthorhombic $a$, $b$, and $c$ axis, as indicated, for crystal 1 (top panel) and crystal 2 (bottom panel). The data for ${\alpha }_b$ in the top panel are taken from Ref. [22]. The dashed lines indicate idealized stepped phase transitions based on an equal-length construction (see text).

Figure 2

Coefficient of linear thermal expansion ${\alpha }_i$, where $i$ is $a$, $b$ or $c$, of UCoGe crystal 2 in longitudinal magnetic fields $B\parallel a,b,c$ as indicated. In panel (a) the arrow indicates the field variation of $T_C$ for $B\parallel a$. In panel (b) the diamonds indicate $T_C\left(B\right)$ for $B\parallel b$.

Figure 3

Coefficient of linear thermal expansion ${\alpha }_b$ of UCoGe crystal 2 in magnetic fields $B\parallel \mathrm{\Delta }{L}_b$ up to 12 T. (Inset) B-T phase diagram. $T_{\mathrm{sc}}$ obtained by tracking the maximum in ${\alpha }_b\left(T\right)$ (magenta points) and by the midpoints of the transition (blue points).

Figure 4

Coefficient of linear thermal expansion of UCoGe crystal 2 in transverse magnetic fields. (a) For $\mathrm{\Delta }{L}_a$ and $B\parallel c$ (up to 0.5 T as indicated). (b) For $\mathrm{\Delta }{L}_c$ and $B\parallel a$ (up to 4 T). (c) For $\mathrm{\Delta }{L}_c$ and $B\parallel b$ (up to 12 T). (c) The curves in field are shifted upwards for clarity. The dashed lines indicate idealized steplike transitions.

Figure 5

B-T phase diagram. $T_{\mathrm{sc}}$ obtained from transverse configuration $B\parallel b\perp \mathrm{\Delta }{L}_c$ by tracking the onset in $\alpha \left(T\right)$ data (red points), idealized transition (blue points), and minimum of the curve (magenta points).

Figure 6

(Top) Coefficient of the volumetric thermal expansion divided by temperature $\beta /T$, as a function of temperature of UCoGe (crystal 2). (Inset) Temperature variation of the relative volume change $\mathrm{\Delta }V/V$ of the same sample. (Bottom) Specific heat divided by temperature $c/T$ as a function of temperature of UCoGe (data taken from Ref. [19]). (Inset) Temperature variation of the Grüneisen parameter $\mathrm{\Gamma }$.

Figure 7

Superconducting (solid circles) and ferromagnetic (stars) $B-T$ phase diagram of UCoGe (crystal 2) plotted as a function of the reduced temperature $T/T_C$, where $T_C$ is the Curie temperature. $T_{\mathrm{sc}}$ and $T_C$ for $B\parallel a,b$, and $c$ are given by red, blue, and green symbols. The blue dashed line represents the function $f\left(B\right)=1-{(B/B_c)}^2$, where $B_c=19.6\mathrm{T}$.