Unconventional critical scaling of magnetization in ferromagnetic uranium superconductors ${\text{UGe}}_2$ and URhGe

We report a dc magnetization study of the critical phenomenon around the ferromagnetic transition temperature $T_{\mathrm{C}}$ in high-quality single crystals of uranium ferromagnetic superconductors ${\text{UGe}}_2$ and URhGe. The critical exponents, $\beta$ for the temperature dependence of the magnetization below $T_{\mathrm{C}}$, $\gamma$ for the magnetic susceptibility, and $\delta$ for the magnetic isothermal at $T_{\mathrm{C}}$, have been determined with a modified Arrott plot, a Kouvel-Fisher plot, and the scaling analysis. Magnetization in the ferromagnetic state has strong uniaxial magnetic anisotropy in the two compounds. However, the universality class of the critical phenomena do not belong to the 3D Ising system. Although the values of $\beta$ in ${\text{UGe}}_2$ and URhGe are close to those in the 3D magnets, the values of $\gamma$ are close to unity, that expected from the mean-field theory. Similar critical exponents have been reported previously for the 3D Ising ferromagnet UIr where superconductivity appears under high pressure. The critical behavior may be limited to a very narrow Ginzburg critical region of $\Delta T_{\mathrm{G}}\sim$ 1 mK because of the strong itinerant character of the $5f$ electrons in the ferromagnetic superconductor UCoGe where the mean-field behavior of the magnetization has been reported. The unconventional critical scaling of magnetization in ${\text{UGe}}_2$, URhGe, and UIr cannot be explained via previous approaches to critical phenomena. The ferromagnetic correlation between the $5f$ electrons differs from that in the 3D Ising system and this difference may be a key point for the understanding of the ferromagnetism where superconductivity emerges.

Figure 1

Modified Arrott plot of magnetization in (a) ${\text{UGe}}_2$ for 48.5 K $\le$ $T$ $\le$ 57.5 K and 0.1 T $\le$ ${\mu }_{0}H$ $\le$ 6.0 T, and in (b) URhGe for 8.6 K $\le$ $T$ $\le$ 10.4 K and 0.1 T $\le$ ${\mu }_{0}H$ $\le$ 2.0 T. Blue lines show fits to the data with Eq. (4).

Figure 2

Magnetic field dependence of magnetization in (a) ${\text{UGe}}_2$ from 48.5 K to 57.5 K and in (b) URhGe from 8.6 K to 10.4 K. Dotted points indicate the critical isotherm at 52.5 K and 9.45 K for ${\text{UGe}}_2$ and URhGe, respectively. Blue lines show fits to Eq. (3) to obtain the critical exponent $\delta$.

Figure 3

Upper panels: Temperature dependence of the spontaneous magnetization $M{}_s\left(T\right)$ and the inverse of the initial magnetic susceptibility ${\chi }^{-1}$ determined from the modified Arrott plot; lower panels: Kouvel-Fisher plots for $M{}_s\left(T\right)$ and ${\chi }^{-1}$ in (a) ${\text{UGe}}_2$ and (b) URhGe.

Figure 4

Effective exponents (a) ${\beta }_{\mathrm{eff}}$ below $T_{\mathrm{C}}$ and (b) ${\gamma }_{\mathrm{eff}}$ above $T_{\mathrm{C}}$ as a function of reduced temperature $t$ [=$|(T-T_{\mathrm{C}})/T_{\mathrm{C}}|$] in ${\text{UGe}}_2$ and URhGe.

Figure 5

Scaled magnetization as a function of renormalized field following Eq. (9) below and above the critical temperature $T_{\mathrm{C}}$ for (a) ${\text{UGe}}_2$ and for (b) URhGe. Solid lines show best-fit polynomials. The magnetization data in the temperature range $t=|(T-T_{\mathrm{C}})/T_{\mathrm{C}}|<0.1$ are shown.