Electronic properties of a heavy-fermion $\mathrm{U}({\mathrm{Ru}}_{0.92}{\mathrm{Rh}}_{0.08}){}_2{\mathrm{Si}}_2$ single crystal

We report the crystal structure and highly anisotropic magnetic, transport, and thermal properties of an exceptionally good single crystal of $\mathrm{U}{\left({\mathrm{Ru}}_{0.92}{\mathrm{Rh}}_{0.08}\right)}_2{\mathrm{Si}}_2$, prepared using a modified Czochralski method. Our study, that also includes neutron diffraction results, shows all the heavy-fermion signatures of pristine ${\mathrm{URu}}_2{\mathrm{Si}}_2$; however, the superconductivity, hidden order, and remnant weak antiferromagnetic orders are absent. Instead, the ground state of the doped system can be classified as a spin liquid that preserves the heavy-fermion character. $\mathrm{U}{\left({\mathrm{Ru}}_{0.92}{\mathrm{Rh}}_{0.08}\right)}_2{\mathrm{Si}}_2$ exhibits a short-range magnetic order distinguished by reflections of a Lorentzian profile at ${\mathrm{q}}_{\mathit{III}}=\left(\frac{1}{2}\frac{1}{2}\frac{1}{2}\right)$ positions that disappear above $\sim 15$ K. The short-range order seems to be a precursor of a long-range magnetic order that occurs with higher Rh concentration. We indicate that these short-range fluctuations involve, at least partially, inelastic scattering processes.

Figure 1

Rocking curves through the 110 nuclear reflection as recorded on E4 at 50 K and 1.7 K. The line segment at the bottom denotes the resolution of the E4 diffractometer at the scattering angle of the 110 reflection. Full lines through the experimental points are the best fits to a Gaussian profile. Note that the profile of the reflection is clearly resolution limited, documenting a high quality of the single crystal.

Figure 2

Plot of the observed versus calculated nuclear structure factors of $\mathrm{U}{\left({\mathrm{Ru}}_{0.92}{\mathrm{Rh}}_{0.08}\right)}_2{\mathrm{Si}}_2$ after the extinction correction (E5 data taken at room temperature). A schematic representation of the ${\mathrm{ThCr}}_2{\mathrm{Si}}_2$ crystal structure adopted by $\mathrm{U}{\left({\mathrm{Ru}}_{1-x}{\mathrm{Rh}}_x\right)}_2{\mathrm{Si}}_2$ is shown in the right inset. The atoms are shown as large (red), intermediate (green), and small (blue) circles that stand for U, Ru/Rh, and Si, respectively. The two single crystals used in the diffraction experiments are shown in the left inset.

Figure 3

Temperature dependence of the static magnetic susceptibility $\chi =M/H$ of the unannealed $\mathrm{U}{\left({\mathrm{Ru}}_{0.92}{\mathrm{Rh}}_{0.08}\right)}_2{\mathrm{Si}}_2$ single crystal measured at 1 T applied along the principal axes. In the inset we show the temperature dependence of the inverse magnetic susceptibility measured in 1 T applied along the tetragonal axis together with the best fit to a Curie-Weiss law to data above 300 K.

Figure 4

The low-temperature detail of the $c$-axis magnetic susceptibilities measured at 1 and 14 T with field applied along the tetragonal axis in zero-field-cooled and field-cooled regimes for the unannealed $\mathrm{U}{\left({\mathrm{Ru}}_{0.92}{\mathrm{Rh}}_{0.08}\right)}_2{\mathrm{Si}}_2$ single crystal. Data obtained at 1 T applied along the $c$ axis on the heat-treated (HT) crystal are shown as well for a comparison. Note an absence of any anomaly in the whole temperature range for all curves.

Figure 5

Field dependence of the $\mathrm{U}{\left({\mathrm{Ru}}_{0.92}{\mathrm{Rh}}_{0.08}\right)}_2{\mathrm{Si}}_2$ single-crystal magnetization measured at various temperatures between 1.8 K and 60 K with field applied along the $a$ and the $c$ axis. Note the nearly linear $M-H$ behavior throughout.

Figure 6

The zero-field low-temperature dependence of the $\mathrm{U}{\left({\mathrm{Ru}}_{0.92}{\mathrm{Rh}}_{0.08}\right)}_2{\mathrm{Si}}_2$ electrical resistivity for current along the $a$ and the $c$ axis together with the best fit described in the main text. In the inset the electrical resistivity measured along the $a$ axis and the $c$ axis in the whole temperature range in zero field and 14 T applied along the tetragonal axis is shown.

Figure 7

The temperature dependence of the net magnetoresistance at 14 T applied along the tetragonal axis and the zero-field state between 2 and 100 K.

Figure 8

The field dependence of the $\mathrm{U}{\left({\mathrm{Ru}}_{0.92}{\mathrm{Rh}}_{0.08}\right)}_2{\mathrm{Si}}_2$ single crystal measured with the current along the $a$ axis (a) and the $c$ axis (b) at various temperatures as a function of field applied along the $c$ axis. Full lines represent best fits to the expression $({\rho }_H$ – ${\rho }_{0T})$/${\rho }_{0T}=aH$ + $b{H}^2$. In the inset of panel (b) we show the temperature dependence of the fit parameters to this formula for both electrical current orientations. In the inset of panel (a) we document a hysteretic behavior for current along the $a$ axis that is found for both current orientations only at 10 and 15 K.

Figure 9

The temperature dependence of the $\mathrm{U}{\left({\mathrm{Ru}}_{0.92}{\mathrm{Rh}}_{0.08}\right)}_2{\mathrm{Si}}_2$ specific heat together with the best fit containing an exponential term (see the main text). In the inset the low-temperature detail of the $C/T\text{vs}{T}^2$ dependence is shown along with the best fits to a linear (solid, blue line) and power-law (dashed, red line) dependencies, respectively.

Figure 10

The temperature dependences of the $\mathrm{U}{\left({\mathrm{Ru}}_{0.92}{\mathrm{Rh}}_{0.08}\right)}_2{\mathrm{Si}}_2$ specific heat measured at different magnetic fields applied along the tetragonal axis. In the inset the magnetic entropy calculated by integration of data shown in the main panel (from which an estimate of a nonmagnetic analog contribution has been subtracted) divided by the temperature up to 14 K is shown.

Figure 11

Diffraction patterns of the $\mathrm{U}{\left({\mathrm{Ru}}_{0.92}{\mathrm{Rh}}_{0.08}\right)}_2{\mathrm{Si}}_2$ single crystal measured at 1.8 K (a) and at 20 K (b) in the $\left(hhl\right)$ orientation using an E2 diffractometer transformed into the reciprocal space. Note that this instrument senses a significant portion of the reciprocal space also above and below the scattering plane. Sharp strong nuclear Bragg reflections of the crystal main phase are observed along with broader features indexable with ${\mathrm{q}}_{III}$ = $\left(\frac{1}{2}\frac{1}{2}\frac{1}{2}\right)$. ${\mathrm{q}}_{III}$ and – ${\mathrm{q}}_{III}$ are shown in (a) by arrows.

Figure 12

Diffraction pattern taken around the $h=\frac{1}{2}k=\frac{1}{2}l=\frac{1}{2}$ position at 1.8 K and at 25 K as measured on the E4 diffractometer. (a) Reciprocal scan along the $\mathrm{[}hh0]$ direction. The scan along the [0 $0l]$ direction is shown in panel (b). Full line through the experimental points are best fits to a Lorentzian profile. The short black line (“H”) at the bottom shows the instrument resolution at the current position.

Figure 13

Temperature dependence of the diffraction intensity taken at the $h=\frac{1}{2}k=\frac{1}{2}l=\frac{1}{2}$ reciprocal space position between 1.7 K and 21 K. In the inset we show an energy scan through this position using the triple-axis spectrometer E1 at 1.7 K and 21 K.