Collinear antiferromagnetic order in $\mathrm{U}{\mathrm{Ru}}_2{\mathrm{Si}}_{2-x}{\mathrm{P}}_x$ revealed by neutron diffraction

The hidden order phase in $\mathrm{U}{\mathrm{Ru}}_2{\mathrm{Si}}_2$ is highly sensitive to electronic doping. A special interest in silicon-to-phosphorus substitution is due to the fact that it may allow one, in part, to isolate the effects of tuning the chemical potential from the complexity of the correlated $f$ and $d$ electronic states. We investigate the new antiferromagnetic phase that is induced in $\mathrm{U}{\mathrm{Ru}}_2{\mathrm{Si}}_{2-x}{\mathrm{P}}_x$ at $x\gtrsim 0.27$. Time-of-flight neutron diffraction of a single crystal ($x=0.28$) reveals $c$-axis collinear ${\mathbf{q}}_{\mathrm{m}}=(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ magnetic order with localized magnetic moments ($\sim 2.1$–$2.6{\mu }_{\mathrm{B}}$). This points to an unexpected analogy between the (Si,P) and (Ru,Rh) substitution series. Through further comparisons with other tuning studies of ${\mathrm{URu}}_2{\mathrm{Si}}_2$, we are able to delineate the mechanisms by which silicon-to-phosphorus substitution affects the system. In particular, both the localization of itinerant $5f$ electrons as well as the choice of ${\mathbf{q}}_m$ appear to be consequences of the increase in chemical potential. Further, enhanced exchange interactions are induced by chemical pressure and lead to magnetic order, in which an increase in interlayer spacing may play a special role.

Figure 1

Simplified phase diagram of $\mathrm{U}{\mathrm{Ru}}_2{\mathrm{Si}}_{2-x}{\mathrm{P}}_x$, adapted from Gallagher et al. [59]. Few percents of phosphorus substitution suppress the hidden order (HO) and, with it, superconductivity (SC). The arrow marks the composition of the long-range antiferromagnetically (AFM) ordered sample investigated in this study.

Figure 2

(a) Overview of Bragg peaks observed at WISH, at 2 K. The upper (lower) panel shows the raw neutron counts detected on the left (right) detector bank, on an arbitrary logarithmic scale. The (101)–$\left(0\overline{1}1\right)$ plane of reciprocal space is indicated by a white dashed line. The (1,1,1) direction, which features the only observed magnetic Bragg peak, is seen below this plane, at a scattering angle of around ${30}^{\circ }$. (b) Perspective view of this data, illustrating the layout of the instrument. (c) Schematic view of the (101)–$\left(0\overline{1}1\right)$ plane of reciprocal space (Note that the magnetic peak at ${\mathbf{q}}_{\mathrm{m}}$ is observed below this plane). The accessible range of momentum transfers is delineated by a broad gray line and peaks seen in (a) are labeled in analogy. The dotted line indicates the momentum transfer at which the magnetic form factor of uranium has decreased by $1/e$ ($\approx$ 5.5 Å${}^{-1}$).

Figure 3

Magnetic susceptibility of the $\mathrm{U}{\mathrm{Ru}}_2{\mathrm{Si}}_{2-x}{\mathrm{P}}_x$ ($x=0.28$) crystal investigated by neutron diffraction, in a field of 0.5 T applied either parallel or perpendicular to the $c$ axis. The right panel shows a detailed view of the $\mathbf{H}\perp \mathbf{c}$ data. As in the parent compound, the characteristics are dominated by the onset of Kondo screening around $T_{\mathrm{coh}}\approx 80$ K, as well as a strong $c$-axis single-ion anisotropy.

Figure 4

Comparison of Bragg intensities calculated for $\mathrm{U}{\mathrm{Ru}}_2{\mathrm{Si}}_{2-x}{\mathrm{P}}_x$ ($x=0.28$) with those measured at WISH at 2 K. The overall scale factor was inferred from a refinement of nuclear intensities (red). The magnitude of the ordered magnetic moment was then fitted separately to reproduce the intensities of magnetic reflections (green). Numerical values of these fits are given in the Supplemental Material [68].

Figure 5

Temperature dependence of the ordered magnetic moment in $\mathrm{U}{\mathrm{Ru}}_2{\mathrm{Si}}_{2-x}{\mathrm{P}}_x$ ($x=0.28$), with an ordering temperature of $T_{\mathrm{N}}=32.6\left(7\right)$ K and critical exponent $\beta =0.31\left(4\right)$. The inset illustrates the emergence of a magnetic Bragg reflection at momentum transfer $\mathbf{Q}={\mathbf{q}}_{\mathrm{m}}=(\frac{1}{2},\frac{1}{2},\frac{1}{2})$.

Figure 6

$c$-axis collinear ${\mathbf{q}}_{\mathrm{m}}=(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ magnetic structure of $\mathrm{U}{\mathrm{Ru}}_2{\mathrm{Si}}_{2-x}{\mathrm{P}}_x$ ($x=0.28$), described by the Shubnikov group $I_c{4}_1/acd$ (No. 142.570). For clarity, only uranium ions are shown and opposite spins are drawn in different colors.

Figure 7

Changes of the lattice parameters $a$ (top) and $c$ (bottom) in the (Ru,Rh) and (Si,P) substitution series. The data are adapted from studies by Burlet et al. [45] and Gallagher et al. [59]. In the lower panel, the critical compositions at which local moment ${\mathbf{q}}_m=(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ magnetic order emerges in either series are marked by arrows.

Figure 8

Pressure dependence of the heavy-fermion coherence scale $T_{\mathrm{coh}}$, inferred from a broad maximum in magnetic susceptibility $\chi \left(T\right)$ curves (cf. Fig. 3). Data for chemical pressure in the present doping series, adapted from Ref. [59], is compared to measurements of the parent compound under applied pressure, reported by Pfleiderer et al. [81]. The arrows and shaded margins indicate the regimes where long-range magnetic order is induced by hydrostatic (red) and chemical (black) pressure.