Influence of hydrostatic pressure on hidden order, the Kondo lattice, and magnetism in ${\mathrm{URu}}_2{\mathrm{Si}}_{2-x}{\mathrm{P}}_x$

Within the chemical substitution series ${\mathrm{URu}}_2{\mathrm{Si}}_{2-x}{\mathrm{P}}_x$, there is an evolution in the ground-state behavior from hidden ordered (HO) for $x\lesssim 0.03$, to Kondo lattice behavior with no ordering (NO) for $0.03\lesssim x\lesssim 0.26$, to antiferromagnetism (AFM-2) for $0.26\lesssim x\lesssim 0.5$ [A. Gallagher et al., Nat. Commun. 7, 10712 (2016); A. Gallagher et al., J. Phys.: Condens. Matter 29, 024004 (2016)]. To better understand what factors control this behavior, temperature-dependent electrical resistivity measurements are performed for this series under applied pressures $P$ up to 20.5 kbar. Specimens in the HO $x$ region show similarities to the parent compound, where HO transforms into antiferromagnetism (AFM-1) at a critical pressure ($P_{\mathrm{c}}$). $P_{\mathrm{c}}$ decreases with increasing $x$ and collapses towards $P=0$ near $x\approx 0.03$, suggesting that AFM-1 occurs at ambient pressure for this concentration. No pressure-induced phase transitions are observed in the NO $x$ region and the AFM-2 state is only weakly suppressed by $P$. Measurements further reveal that AFM-1 and AFM-2 are distinct from each other. Calculations of the wave functions using the tight-binding Hartree-Fock approximation are performed and show (i) that the radial probability distributions for the phosphorus ions are more tightly bound than those for the silicon and (ii) that the energy difference between the orbitals decreases with increasing $x$. The cumulative effect of these two factors is that $\mathrm{Si}\to \mathrm{P}$ substitution decreases the hybridization strength, which correlates with the weakening of HO. At large $x$, additional effects such as electrical charge tuning also play an important role in determining the ground-state behavior.

Figure 1

The room-temperature-normalized electrical resistivity $\rho \left(T\right)/\rho$(300 K) for ${\mathrm{URu}}_2{\mathrm{Si}}_{2-x}{\mathrm{P}}_x$ at ambient pressure for select concentrations in the hidden order, no order, and antiferromagnetic $x$ regions ($x=0$, 0.03, 0.1, and 0.35). Arrows indicate the transition temperature $T_0$ for $x=0$ and 0.03 and $T_{\mathrm{N}2}$ for $x\approx 0.35$. Inset: Transition temperature and chemical concentration $T\text{−}x$ phase diagram from Ref. [48].

Figure 2

The temperature-dependent electrical resistivity $\rho \left(T\right)$ at various pressures $P$ for ${\mathrm{URu}}_2{\mathrm{Si}}_{2-x}{\mathrm{P}}_x$: (a) $x=0$, (d) $x\approx 0.01$, (g) $x\approx 0.02$, (j) $x\approx 0.03$, (m) $x\approx 0.2$, and (p) $x\approx 0.35$. The samples $x\approx 0.1$ and 0.33 for the NO and AFM-2 regions, respectively, were measured and their results are similar to those for $x\approx 0.2$ and 0.35, therefore they are not shown. Black arrows indicate the direction of applied pressure. Insets: The derivative of the data with respect to temperature $\partial \rho /\partial T$ zoomed in on the HO transition (b, e, h, k), the low-$T$ region of no ordering (n), and the AFM-2 transition (q). For clarity, several measured pressure curves have been omitted.

Figure 3

(a,b) A zoom-in on the electrical resistivity $\rho$ vs temperature $T$ for ${\mathrm{URu}}_2{\mathrm{Si}}_{2-x}{\mathrm{P}}_x$ in the vicinity of the ordering temperature for $x\approx 0.01$ and 0.02, respectively. (c) The pressure dependence of the peak height $\mathrm{\Delta }\rho /{\rho }_{\min }$ as defined in (a) for $x=0$, 0.01, and 0.02. Inset: Temperature width $W$ at the phase transition. Results for $x\approx 0.01$ are similar to those for $x=0$ but are omitted for clarity. Lines are guides for the eye, arrows indicate the critical pressure $P_{\mathrm{c}}$ for the $\mathrm{HO}\to \mathrm{AFM}$-1 transition, and spanning bars represent the range of pressure where $P_{\mathrm{c}}$ could be defined based on the method described in the text.

Figure 4

Transition temperature $T_{0,\mathrm{N}1}$ vs phosphorus concentration $x$ vs applied pressure $P$ phase diagram for ${\mathrm{URu}}_2{\mathrm{Si}}_{2-x}{\mathrm{P}}_x$ for $x=0$, 0.01, 0.02, and 0.03 with linear fits (solid lines). The critical pressure $P_{\mathrm{c}}$ for the $\mathrm{HO}\to \mathrm{AFM}$-1 transition is indicated by the dashed vertical lines and gray circles with spanning bars [the same as in Fig. 3], and the solid gray line is a linear fit to $P_{\mathrm{c}}$ that marks the HO/AFM-1 phase boundary. Inset: The energy gap parameter $\mathrm{\Delta }$, extracted from fits to data with Eq. (1) as described in the text, for these concentrations and for $x\approx 0.35$. The error bars, determined from the fitting algorithm, are smaller than the symbols, except for $x\approx 0.03$.

Figure 5

The radial probability distributions for U, Si, and P expressed in atomic units.