Intriguing behavior of $\mathrm{UC}{\mathrm{o}}_{1-x}\mathrm{R}{\mathrm{h}}_x\mathrm{Ge}$ ferromagnets in magnetic field along the $b$ axis

Single crystals of solid solutions of ferromagnetic superconductors UCoGe and URhGe were investigated by magnetization and specific-heat measurements. These compounds behave as anisotropic ferromagnets with the easy-magnetization direction along the orthorhombic $c$ axis. The maximum Curie temperature $T_C=20\mathrm{K}$ has been observed in $\mathrm{UC}{\mathrm{o}}_{1-x}\mathrm{R}{\mathrm{h}}_x\mathrm{Ge}$ for $x=0.4$. The main interest of the study was the intriguing behavior in magnetic fields applied along the $b$ axis. The temperature dependence of the $b$-axis magnetic susceptibility exhibits a maximum at a characteristic temperature $T_{\max }$. For low Rh concentrations $\left(x<0.4\right)T_{\max }>T_C$ and the magnetic entropy $S_{\mathrm{mag}}$ at $T_C$ is very low, whereas $T_{\max }=T_C$ and the $S_{\mathrm{mag}}$ at $T_C$ is much higher for $x\ge 0.4$. A metamagneticlike anomaly in magnetization is observed in all compounds, at a critical field which appears on a similar energy scale with the value of the corresponding $T_{\max }$. The character of three paramagnetic regimes which can be distinguished in the T-$x$ and H-$x$ phase diagrams, normal, correlated, and field polarized, is discussed in relation to the field-induced reentrant superconductivity in the parent compounds.

Figure 1

(a) Thermomagnetic curves and (b) magnetization isotherms of $\mathrm{UC}{\mathrm{o}}_{1-x}\mathrm{R}{\mathrm{h}}_x\mathrm{Ge}$ compounds in fields applied along the $c$ axis. Magnetization data of UCoGe are taken from Ref. [3] and URhGe from Ref. [29]. The arrows in (a) indicate $T_C$.

Figure 2

Temperature dependencies of magnetization of $\mathrm{UC}{\mathrm{o}}_{1-x}\mathrm{R}{\mathrm{h}}_x\mathrm{Ge}$ compounds for several values of magnetic field applied along the $b$ axis. Figures show the evolution of the $T_C$ and $T_{\max }$ throughout the $\mathrm{UC}{\mathrm{o}}_{1-x}\mathrm{R}{\mathrm{h}}_x\mathrm{Ge}$ system. Panels (a)–(e) show curves in a magnetic field of 3 T where the anomaly at $T_{\max }$ is visible. The estimation of $T_{\max }$ is difficult in $\mathrm{UC}{\mathrm{o}}_{0.7}\mathrm{R}{\mathrm{h}}_{0.3}\mathrm{Ge}$; $T_{\max }$ seems to keep a similar value as $\mathrm{UC}{\mathrm{o}}_{0.8}\mathrm{R}{\mathrm{h}}_{0.2}\mathrm{Ge}$ but overlaid with the ferromagnetic signal due to an increasing $T_C$. Panels (f)–(j) show all recorded curves in various magnetic fields. Panels (f) and (i) representing $\mathrm{UC}{\mathrm{o}}_{0.6}\mathrm{R}{\mathrm{h}}_{0.4}\mathrm{Ge}$ and $\mathrm{UC}{\mathrm{o}}_{0.9}\mathrm{R}{\mathrm{h}}_{0.1}\mathrm{Ge}$ are displayed in full temperature scale as representative examples of $T_C=T_{\max }$ and $T_C<T_{\max }$ behavior.

Figure 3

Magnetization isotherms of all $\mathrm{UC}{\mathrm{o}}_{1-x}\mathrm{R}{\mathrm{h}}_x\mathrm{Ge}$ compounds measured in the $b$-axis pulsed field; data for parent compounds are identical to those reported in Ref. [3] and displayed with kind permission of authors W. Knafo and D. Aoki. The inset shows development of the magnetization slopes dM/dH at $M_{\mathrm{ins}}$ (see also Table 1). The dashed lines indicate the onset $M_{\mathrm{ins}}$, inflection $M_{\inf }$, and the end $M_{\mathrm{end}}$ of the metamagnetic anomalies. The absolute values of the pulsed-field magnetization data were calibrated using precise superconducting quantum interference device magnetization data represented as dots in the magnetic field interval 0–7 T.

Figure 4

The temperature dependence of specific heat ($C_p/T$ vs T plots) of $\mathrm{UC}{\mathrm{o}}_{1-x}\mathrm{R}{\mathrm{h}}_x\mathrm{Ge}$ compounds for (a) $x<0.4$ (with $T_C<T_{\max }$) and (b) $x\ge 0.4$ (with $T_C=T_{\max }$). The arrows indicate $T_C$. Data for the parent compounds UCoGe and URhGe have been taken from Refs. [26] and [32], respectively.

Figure 5

The temperature dependence of specific heat and its components (C/T vs T plots) and magnetic entropy $S_{\mathrm{mag}}$ for (a) $x=0.1$ and (b) $x=0.8$. The value of $S_{\mathrm{mag}}$ is also expressed in the unit Rln2 at the red horizontal hairline.

Figure 6

(a) T-$x$ and (b) H-$x$ phase diagrams of the $\mathrm{UC}{\mathrm{o}}_{1-x}\mathrm{R}{\mathrm{h}}_x\mathrm{Ge}$ system. The value of $T_{\max }$, $H_{\mathrm{R}}$, and $H_{\mathrm{m}}$ for URhGe and UCoGe, respectively, were taken from Ref. [3]. The blue point in panel (b) marks the critical field of the ferromagnetic phase boundary in UCoGe; the point was taken from Ref. [11]. The long-dashed line in the H-$x$ diagram tentatively indicates the evolution of FM → CPM crossover at $H_{\mathrm{b}}$. The grey line in the T-$x$ diagrams in panel (a) tentatively separates the low- and high-$S_{\mathrm{mag}}$ concentration regions.

Figure 7

(a) The concentration dependencies of (a) the $T_{\max }/H_{\mathrm{m}}$ and $T_C/H_{\mathrm{m}}$ ratios and (b) the measured $H_{\mathrm{m}}$ and calculated $H_{\mathrm{b}}$ characteristic fields of metamagneticlike anomalies for $\mathrm{UC}{\mathrm{o}}_{1-x}\mathrm{R}{\mathrm{h}}_x\mathrm{Ge}$ compounds.

Figure 8

The concentration dependence of the average slope of the magnetization curve in a field along the easy magnetization axis ($c$ axis). The slope was calculated for the magnetic field interval 1–7 T. The point for URhGe was calculated from data in Ref. [29] and points for UCoGe from data in Ref. [3] [green point: pulsed-field data in Fig. 1] and Ref. [43] (blue point: steady-field data in the interval 1–5 T). Two different values for UCoGe are caused by a strong sample dependence of FM parameters [25, 26] but also different sweep rate of magnetic field.

Figure 9

The evolution of the parameters $S_{\mathrm{mag}}$, γ, and height of peak $\mathrm{\Delta }{C}_{\mathrm{p}}/T$ at ferromagnetic transitions for all compositions in $\mathrm{UC}{\mathrm{o}}_{1-x}\mathrm{R}{\mathrm{h}}_x\mathrm{Ge}$.

Figure 10

The magnetization loops along all three crystallographic axes of all alloy compounds in the $\mathrm{UC}{\mathrm{o}}_{1-x}\mathrm{R}{\mathrm{h}}_x\mathrm{Ge}$ system for $x=0.1$ (a), 0.2 (b), 0.3 (c), 0.4 (d), and 0.8 (f). Panels are displayed in the identical magnetization scale to enhance the evolution of values of spontaneous $\left({\mu }_{\mathrm{s}}\right)$ and saturated magnetizations.

Figure 11

Specific-heat data of all compounds in $\mathrm{UC}{\mathrm{o}}_{1-x}\mathrm{R}{\mathrm{h}}_x\mathrm{Ge}$ system with the evaluated values of magnetic entropy $S_{\mathrm{mag}}$ for $x=0.2$ (a), 0.3 (b), 0.4 (c), and 0.9 (d). The value of $S_{\mathrm{mag}}$ is also displayed in the Rln2 scale and marked by the red horizontal hairline. A general polynomial function with dominant quadratic term was used for composition with $T_C>15\mathrm{K}$ where the simple Debye $\sim T^2$ dependence does not provide usable result to extract the phonon contribution.

Figure 12

Lattice parameters throughout the $\mathrm{UC}{\mathrm{o}}_{1-x}\mathrm{R}{\mathrm{h}}_x\mathrm{Ge}$ system; error bars are smaller than the size of points.