Two magnetic Grüneisen parameters in the ferromagnetic superconductor ${\text{UGe}}_2$

We report ambient-pressure magnetization, heat capacity, and thermal-expansion measurements of the ferromagnetic superconductor ${\text{UGe}}_2$ in high magnetic fields. An analysis of the magnetic heat capacity derived from both magnetization and specific-heat data shows that ${\text{UGe}}_2$ is well described in the framework of the molecular-field theory. Our heat-capacity and thermal-expansion results reveal a clear crossover regime, a feature that illustrates the proximity to the quantum critical end point of a first-order boundary between two different ferromagnetic phases. Furthermore, we show that the ferromagnetic contribution to these thermodynamic quantities can be split into two terms with distinct Grüneisen parameters.

Figure 1

(Color online) The schematic $\left(p,T\right)$ phase diagram of ${\text{UGe}}_2$ (Ref. 13). Thick lines represent first-order transitions and thin lines denote second-order transitions. The dashed line indicates a crossover while the dots mark the positions of critical points. The superconducting region is represented in red (black area at bottom).

Figure 2

(Color online) (a) Temperature dependence of the magnetization $M\left(T\right)$ of ${\text{UGe}}_2$ for fields up to 8 T. (b) $M^2$ as a function of $T^2$ for temperatures below 20 K. The dashed line is a fit to the Stoner expression $M=M_0\left(1-\alpha T^{3/2}{e}^{-\Delta /T}\right)$, using the gap value, $\Delta =40\text{K}$, derived from neutron measurements (Ref. 14).

Figure 3

(Color online) Temperature dependence of $C/T$ of ${\text{UGe}}_2$ for several fields. The lattice contribution (full line, see Sec. 3B2) is also represented. The inset illustrates the field dependence of the low-temperature linear term $\gamma$.

Figure 4

(Color online) (a) Magnetic and electronic heat capacity obtained by subtracting the lattice contribution $C_{\text{lat}}/T$ (see Sec. 3B2) from the measured heat capacity $C/T$. (b) Magnetic specific heat $C_{\text{MFA}}/T$ calculated in the MFA by applying Eq. (1) to the magnetization data.

Figure 5

(Color online) Temperature dependence of $\left(C-C_{\text{MFA}}\right)/T$ for several magnetic fields. The lattice fit (full line) is also represented together with its Debye (dashed line) and Einstein components (dotted and dash-dotted lines).

Figure 6

(Color online) Temperature dependence of the coefficients of linear thermal expansion along the $a$, $b$, and $c$ axes and volume for fields up to 8 T. The anomaly at $T_x$ is clearly visible. The lattice contribution for each direction is also shown (dashed line).

Figure 7

(Color online) Temperature dependence of the volume magnetic thermal expansion ${\alpha }_v^{\text{mag}}\left(T\right)$ for fields up to 8 T.

Figure 8

(Color online) Temperature dependence of the volume magnetic thermal expansion ${\alpha }_v^{\text{mag}}\left(T\right)$ (full lines) together with the scaled magnetic specific heat (dashed lines) for 0 and 8 T. We observe that ${\alpha }_v^{\text{mag}}\left(T\right)$ is proportional to $C_{\text{MFA}}\left(T\right)$ only in a restricted temperature range below $T_x$.

Figure 9

(Color online) Temperature dependence of the magnetic heat capacity (a) and magnetic thermal expansion (b) in zero field. These two quantities are decomposed into two contributions related to FM1 (red) and FM2 (blue).

Figure 10

(Color online) Temperature dependence of the ambient-pressure FM1 magnetic thermal expansion. It is compared to the thermal expansion measured along the $b$ axis at 13 kbar (Ref. 27). Because of its small lattice contribution, ${\alpha }_b\left(T\right)$ is almost entirely of magnetic origin.