Thermal expansion under uniaxial pressure in URu${}_2$Si${}_2$

The temperature dependence of the thermal expansion coefficient $\alpha$ has been measured in URu${}_2$Si${}_2$ under uniaxial pressure $\sigma$ along the [100] and [110] directions up to $\sim 0.6$ GPa. For both cases, the hidden-order phase transition temperature $T_0$ increases with increasing uniaxial pressure. An anomaly $\Delta \alpha$ of the thermal expansion coefficient at $T_0$ appears to be enhanced under a small uniaxial pressure ($<0.03$ GPa). An antiferromagnetically ordered phase appears at $T_X$ above the critical pressure ${\sigma }_X\sim 0.25$ GPa for uniaxial pressure along the [100] direction, whereas no antiferromagnetic transition is detected up to $\sim 0.6$ GPa for uniaxial pressure along the [110] direction, indicating that the critical pressure is beyond $\sim 0.6$ GPa for the [110] case. In the previously reported hydrostatic case, an antiferromagnetic phase transition appears above the critical pressure $P_X\sim 0.5$ GPa. The difference in critical pressure between ${\sigma }_X$ and $P_X$ can be understood in terms of the pressure dependence of the U-U distance in the basal (001) plane.

Figure 1

Sample configuration for thermal expansion measurements along the three directions under uniaxial pressure ${\sigma }_{100}$ along the [100] direction. Three strain gauges are attached to the surfaces of the single crystal sample: (001) planes for ${\alpha }_{100}$ and ${\alpha }_{010}$, (010) plane for ${\alpha }_{001}$. For the case of ${\sigma }_{110}$, three strain gauges are attached to the surfaces: (001) planes for ${\alpha }_{110}$ and ${\alpha }_{1\overline{1}0}$, (110) plane for ${\alpha }_{001*}$.

Figure 2

$T$ dependence of thermal expansion ${\alpha }_{010}$ under uniaxial pressure ${\sigma }_{100}$. For clarity a suitable offset of $\alpha$ is inserted for each value of ${\sigma }_{100}$. Black arrows indicate the ${\sigma }_{100}$ dependence of $T_X$. $\Delta {\alpha }_{010}$ and $\delta {\alpha }_{010}$ indicate the anomalies of ${\alpha }_{010}$ at $T_0$ and $T_X$, respectively. $\Delta {\alpha }_{010}$ at $\sigma >0$ is $2\sim 3$ times larger than $\Delta {\alpha }_{010}\left(0\right)$ at $\sigma =0$. $\delta {\alpha }_{010}$ increases with increasing ${\sigma }_{100}$.

Figure 3

The $T$ dependencies of thermal expansion parameters ${\alpha }_{100}$, ${\alpha }_{010}$, and ${\alpha }_{001}$ are shown for ${\sigma }_{100}=0.53$ GPa. Plots of the three $\alpha$ are displaced vertically for clarity. For ${\alpha }_{100}$ and ${\alpha }_{001}$, no clear anomaly is detected at $T_X$, whereas a clear anomaly for these $\alpha$ is observed at $T_0$ similar to that of ${\alpha }_{010}$. $\Delta {\alpha }_{100}$, and $\Delta {\alpha }_{001}$ are enhanced suddenly under a small uniaxial pressure and become almost constant, again in similar fashion to $\Delta {\alpha }_{010}$.

Figure 4

Phase diagram under uniaxial pressure ${\sigma }_{100}$ along the [100] direction. For comparison, the hydrostatic pressure $P$ dependencies of $T_0$ and $T_X$ are also presented (open symbols and blue solid line).[8] It should be noted that the difference between the $P$ and ${\sigma }_{100}$ dependence of $T_0$ cannot be clearly shown in this figure (see Fig. 8). The ${\sigma }_{100}$ dependence of $T_X$ with an offset of $\sim 0.25$ GPa is similar to the $P$ dependence of $T_X$. The dashed line is the expected ${\sigma }_X\sim 0.25$ GPa and ${\sigma }_{100}$ dependence of $T_X$, which needs to be verified in another measurement.

Figure 5

$T$ dependence of the thermal expansion ${\alpha }_{1\overline{1}0}$ under uniaxial pressure ${\sigma }_{110}$ along the [110] direction. A suitable offset of $\alpha$ is added for each pressure value. The antiferromagnetic transition $T_X$ is not detected, although the hidden-order transition $T_0$ is observed. The anomaly around 12 K at 0.6 GPa is considered to be experimental noise.

Figure 6

$T$ dependencies of the thermal expansions ${\alpha }_{1\overline{1}0}$, ${\alpha }_{110}$, and ${\alpha }_{001*}$ under uniaxial pressure ${\sigma }_{110}=0.48$ GPa along the [110] direction. Suitable offsets of $\alpha$ have been inserted for clarity. The antiferromagnetic transition $T_X$ is not detected for any of the three directions. Similar results were obtained at other pressures.

Figure 7

Phase diagram under uniaxial pressure ${\sigma }_{110}$ along the [110] direction. No antiferromagnetic phase transition is detected up to $\sim 0.6$ GPa down to 8 K.

Figure 8

${\sigma }_i$ dependence of the hidden-order transition $T_0$ using results presented in Figs. 4 and 7. The solid lines are obtained using linear least–squares fits. $d{T}_0/d{\sigma }_i$ seems to be slightly larger for ${\sigma }_{110}$. The expected slope $d{T}_0/d{\sigma }_i=0.8$ K/GPa at ${\sigma }_i=0$ from the Ehrenfest relation is presented as a green dashed line. A crossover from this slope to the slope of the red and blue lines is considered to take place below $\sim$0.03 GPa. The hydrostatic $P$ dependence of $T_0$ is presented (triangles)[8] for comparison.

Figure 9

Schematic description of the basal (001) plane in tetragonal URu${}_2$Si${}_2$. The U-U distance is defined as the shortest bonding length between U atoms in the basal plane, which is indicated by red arrows. In the case of ${\sigma }_{100}$ there are two bonding lengths, i.e., parallel (shortest) and perpendicular to ${\sigma }_{100}$. In contrast, in the case of ${\sigma }_{110}$ there is a unique bonding length. As shown in Fig. 10, the reduction of U-U distance is less effective with ${\sigma }_{110}$.

Figure 10

Schematic pressure dependence of the U-U distance in the basal plane.[9] When the shortest U-U distance is reduced to the relevant critical value by any type of pressure, a first-order transition from hidden order to an AFM ordered state takes place.