Wing structure in the phase diagram of the Ising ferromagnet URhGe close to its tricritical point investigated by angle-resolved magnetization measurements

High-precision angle-resolved dc magnetization and magnetic torque studies were performed on a single-crystalline sample of URhGe, an orthorhombic Ising ferromagnet with the $c$ axis being the magnetization easy axis, in order to investigate the phase diagram around the ferromagnetic (FM) reorientation transition in a magnetic field near the $b$ axis. We have clearly detected a first-order transition in both the magnetization and the magnetic torque at low temperatures, and determined detailed profiles of the wing structure of the three-dimensional $T-H_b-H_c$ phase diagram, where $H_c$ and $H_b$ denote the field components along the $c$ and the $b$ axes, respectively. The quantum wing critical points are located at ${\mu }_0{H}_c\sim \pm 1.1$ T and ${\mu }_0{H}_b\sim 13.5$ T. Two second-order transition lines at the boundaries of the wing planes rapidly tend to approach each other with increasing temperature up to $\sim 3$ K. Just at the zero conjugate field ($H_c=0$), however, a signature of the first-order transition can still be seen in the field derivative of the magnetization at $\sim 4$ K, indicating that the tricritical point exists in a rather high temperature region above 4 K. This feature of the wing plane structure is consistent with the theoretical expectation that three second-order transition lines merge tangentially at the tricritical point.

Figure 1

Example of the raw capacitance ($C$) data (solid circles) obtained at 0.21 K in magnetic fields tilted by $1.4^{\circ }$ from the $b$ axis in the $bc$ plane, with the field gradient $G=0$ and 8 T/m. Taking a difference of these two yields the magnetization curve (solid squares).

Figure 2

Schematic view of the two-axis rotation device. $\theta$ and $\varphi$ rotations are performed by a home-made tilting stage and a piezo-stepper-driven goniometer, respectively. The angle of the tilting stage is controlled from the top of the insert by a screw rod that is thermally isolated.

Figure 3

The raw capacitance data $C$ near $H_{\mathrm{R}}\left(\theta \right)$ with zero field gradient, measured at 0.5 K for several $\theta$ values, where $\theta$ is the angle between $H$ and the $b$ axis in the $bc$ plane. The black dotted line is the zero torque state. These data were collected in a different run from the one in Fig. 1.

Figure 4

The angular $\theta$ dependence of the capacitance $C$ with two different field gradient values, $G=0$ and 8 T/m, obtained at 0.5 K in 10.5 T below $H_{\mathrm{R}}$. In the yellow hatched region, the torque component changes so dramatically with $\theta$ that the precise evaluation of the magnetization becomes difficult. The inset schematically shows the $\theta$ evolution of the FM domains with positive and negative components of $M_c$ (solid arrows) in a magnetic field $H$ (open arrows). There is no bulk magnetization $M_c$ at $\theta =0^{\circ }$, and the zero-torque ($\tau =0$) state persists irrespective of the magnitude of $H_b$.

Figure 5

Magnetic torque divided by field, $\mathrm{\Delta }{C}_{\tau }/H$, of URhGe measured at (a) 0.25, (b) 3, and (c) 6 K in fields near $H_{\mathrm{R}}\left(\theta \right)$ with $\theta =0^{\circ }, 0.{79}^{\circ }, 1.{65}^{\circ }, 3.{64}^{\circ }$, and $5.{64}^{\circ }$, together with the differential curves $d\left(\mathrm{\Delta }{C}_{\tau }\right)/dH$ [(d)–(f)].

Figure 6

(a) $\tilde{M}\left(H\right)$ of URhGe measured in the magnetic field along the $b$ axis. (b) The field derivative $d\tilde{M}/dH$ of the magnetization curves in (a). In these figures, only the down-sweep traces are plotted for simplicity. The inset of (b) shows $d\tilde{M}/dH$ at $T=0.25$ K and $\theta =0^{\circ }$ for both up- and down-field sweeps.

Figure 7

(a) Temperature dependence of $H_{\mathrm{R}}\left(\theta \right)$ obtained from the present measurements at $\theta =0^{\circ }, 1.{65}^{\circ }$, and $3.{64}^{\circ }$, together with the results of the previous study (dashed line) [14]. The inset is an expanded plot for $\theta =0^{\circ }$, indicating $H_{\mathrm{R}}\left(T\right)$ defined at up-sweep (open squares) and down-sweep (open circles) fields. (b) Temperature evolution of the peak amplitude of $d\tilde{M}/dH$ at $H_{\mathrm{R}}$, obtained at several $\theta$. (c) Temperature evolution of the transition width, obtained at several $\theta$.

Figure 8

The magnetization curves $M\left(H\right)$ of URhGe near $H_{\mathrm{R}}\left(\theta \right)$ ($\theta =0.{79}^{\circ }, 1.{65}^{\circ }, 3.{64}^{\circ }$, and $5.{64}^{\circ }$), measured at (a) 0.25, (b) 3, and (c) 6 K, together with their differential curves $dM/dH$ for (d) 0.25, (e) 3, and (f) 6 K. For comparison, $\tilde{M}\left(H\right)$ and $d\tilde{M}/dH$ at $\theta =0^{\circ }$ are also plotted. The insets in (a) and (b) show the angular variation of the magnetization jump $\mathrm{\Delta }M$ ($\theta \ge 0.{79}^{\circ }$, solid squares) measured at 0.25 and 3 K, respectively. The solid circles are the linear extrapolation of $\mathrm{\Delta }M$ to $\theta =0^{\circ }$, which are a factor of 0.7 smaller than $\mathrm{\Delta }\tilde{M}$ measured at $\theta =0^{\circ }$.

Figure 9

The contour plot of $dM/dH$ near the FM wing structure of URhGe in the $H_b-H_c$ plane for temperatures 0.25, 1, 2, 3, 4.2, and 6 K. Green dots represent the peak position of $dM/dH$ obtained at $\theta$ = $0^{\circ }, 0.{79}^{\circ }, 1.{65}^{\circ }, 3.{64}^{\circ }$, and $5.{64}^{\circ }$, and the green solid lines are guides to the eye. Mirrored copy data are plotted for $\theta <0^{\circ }$.

Figure 10

The contour plot of $\left|d\right(\mathrm{\Delta }{C}_{\tau })/dH|$ near the FM wing structure of URhGe in the $H_b-H_c$ plane for temperatures 0.25, 1, 2, 3, 4.2, and 6 K. Green dots represent the peak position of $\left|d\right(\mathrm{\Delta }{C}_{\tau })/dH|$ obtained at $\theta$ = $0^{\circ }, 0.{79}^{\circ }, 1.{65}^{\circ }, 3.{64}^{\circ }$, and $5.{64}^{\circ }$, and the green solid lines are guides to the eye. Mirrored copy data are plotted for $\theta <0^{\circ }$.

Figure 11

The color contour plot of the peak amplitude of (a) $dM/dH$ and (b) $\left|d\right(\mathrm{\Delta }{C}_{\tau })/dH|$, projected on the $T-H_c$ plane. These plots give imaging of the wing plane, viewed from the $H_b$ axis. These plots are constructed from the data obtained at $T=0.5$, 1.5, 2.5, 3.5, and 5 K (not shown), in addition to those given in Figs. 9 and 10. The white dots in these figures show the data points from which the color mappings are generated. A schematic $T-H_b-H_c$ phase diagram is given in (c).