Unstable spin-ice order in the stuffed metallic pyrochlore ${\mathrm{Pr}}_{2+x}{\mathrm{Ir}}_{2-x}{\mathrm{O}}_{7-\delta }$

Specific heat, elastic neutron scattering, and muon spin rotation $\left(\mu \mathrm{SR}\right)$ experiments have been carried out on a well-characterized sample of “stuffed” (Pr-rich) ${\mathrm{Pr}}_{2+x}{\mathrm{Ir}}_{2-x}{\mathrm{O}}_{7-\delta }$. Elastic neutron scattering shows the onset of long-range spin-ice “2-in/2-out” magnetic order at $T_M=0.93$ K, with an ordered moment of $1.7\left(1\right){\mu }_{\mathrm{B}}/\text{Pr}$ ion at low temperatures. Approximate lower bounds on the correlation length and correlation time in the ordered state are 170 Å and 0.7 ns, respectively. $\mu \mathrm{SR}$ experiments yield an upper bound 2.6(7) mT on the local field $B_{\mathrm{loc}}^{4f}$ at the muon site, which is nearly two orders of magnitude smaller than the expected dipolar field for long-range spin-ice ordering of $1.7{\mu }_B$ moments (120–270 mT, depending on muon site). This shortfall is due in part to splitting of the non-Kramers crystal-field ground-state doublets of near-neighbor ${\mathrm{Pr}}^{3+}$ ions by the ${\mu }^{+}$-induced lattice distortion. For this to be the only effect, however, $\sim 160$ Pr moments out to a distance of $\sim 14$ Å must be suppressed. An alternative scenario, which is consistent with the observed reduced nuclear hyperfine Schottky anomaly in the specific heat, invokes slow correlated Pr-moment fluctuations in the ordered state that average $B_{\mathrm{loc}}^{4f}$ on the $\mu \mathrm{SR}$ time scale $(\sim {10}^{-7}\text{s})$, but are static on the time scale of the elastic neutron scattering experiments $(\sim {10}^{-9}\text{s})$. In this picture, the dynamic muon relaxation suggests a ${\mathrm{Pr}}^{3+}4f$ correlation time of a few nanoseconds, which should be observable in a neutron spin echo experiment.

Figure 1

Temperature dependence of the specific heat of ${\mathrm{Pr}}_{2+x}{\mathrm{Ir}}_{2-x}{\mathrm{O}}_{7-\delta }$ in zero field. Filled circles: experimental total specific heat. Dashed curve: calculated specific heat due to a reduced nuclear Schottky anomaly (see text). Open circles: specific heat after subtracting the nuclear Schottky anomaly from the total specific heat.

Figure 2

Temperature dependence of elastic neutron scattering intensity of ${\mathrm{Pr}}_{2+x}{\mathrm{Ir}}_{2-x}{\mathrm{O}}_{7-\delta }$ at the position of the ${\mathbf{q}}_m=\left(100\right)$ reflection. The intensity measured at $T=2$ K was subtracted as a background. Curve: Ising mean-field theory fit to the data, which yields a transition temperature of $T_M=0.93\left(1\right)$ K. (Inset) Sketch of the 2-in/2-out magnetic structure.

Figure 3

Elastic and quasielastic neutron scattering intensity of ${\mathrm{Pr}}_{2+x}{\mathrm{Ir}}_{2-x}{\mathrm{O}}_{7-\delta }$ measured at 0.3 K on CNCS, $T=1.7$ K data subtracted. See text for definitions. (a) Scattering intensity as a function of momentum and energy, $\lambda =7.26$ Å. (b) Momentum dependence of the energy-integrated $(\left|E\right|<0.03$ meV) intensity at $T=0.3$ K. Curve: fit to set of Voigt profiles plus a polynomial background. (c) Energy dependence at three Bragg positions, $\lambda =9.04$ Å. Solid curves: fits to Voigt profiles. Dashed curves: associated Lorentzian broadening.

Figure 4

Relaxation of ${\mu }^{+}$ spin polarization $P\left(t\right)$ in ${\mathrm{Pr}}_{2+x}{\mathrm{Ir}}_{2-x}{\mathrm{O}}_{7-\delta }$ at $T=1.0$ K and weak longitudinal field ${\mu }_0{H}_L=0.39$ mT, obtained from asymmetry data. (a) EDSKT fit [Eq. (3)], ${\chi }_{\mathrm{red}}^2=1.187$. (b) DKT fit (Ref. [14]), ${\chi }_{\mathrm{red}}^2=1.191$.

Figure 5

Relaxation of ${\mu }^{+}$ spin polarization $P\left(t\right)$ in ${\mathrm{Pr}}_{2+x}{\mathrm{Ir}}_{2-x}{\mathrm{O}}_{7-\delta }$ at various longitudinal fields $H_L,T=0.1$ K. Curves: EDSKT fits.

Figure 6

Temperature dependence of (a) quasistatic ${\mu }^{+}$ relaxation rate $\mathrm{\Delta }$ and (b) initial $\left(t=0\right)$ sample asymmetry $A_0(1-f_{\mathrm{Ag}})$ (Sec. 2d) from EDSKT fits to weak-LF $\mu \mathrm{SR}$ data from ${\mathrm{Pr}}_{2+x}{\mathrm{Ir}}_{2-x}{\mathrm{O}}_{7-\delta }$. Inset to (a): data in the vicinity of the magnetic ordering temperature $T_M=0.93$ K from elastic neutron scattering.

Figure 7

Lattice sums for the ${\mu }^{+}$-induced suppressed-moment hypothesis in ${\mathrm{Pr}}_{2+x}{\mathrm{Ir}}_{2-x}{\mathrm{O}}_{7-\delta }$ as functions of suppression radius $r_s$ at three assumed ${\mu }^{+}$ sites (see text). Triangles: number $n_s$ of suppressed-moment sites. Circles: ${\mathrm{Pr}}^{3+}4f$ dipolar field $B_{\mathrm{loc}}^{4f}$ at ${\mu }^{+}$ site assuming the ordered spin-ice structure. Solid lines: ${\rho }_{\mathrm{Pr}}{V}_s$, where ${\rho }_{\mathrm{Pr}}$ is the Pr-site number density and $V_s=(4\pi /3)r_s^3$ is the suppression volume. Dashed lines: $8{\mu }_{\mathrm{B}}/r_s^{-3}$ (see text).

Figure 8

Temperature dependence of ${\mu }^{+}$ dynamic relaxation parameters in ${\mathrm{Pr}}_{2+x}{\mathrm{Ir}}_{2-x}{\mathrm{O}}_{7-\delta }$ from fits to weak-LF $\mu \mathrm{SR}$ data. Circles: $\lambda \left(T\right)$ from EDSKT fits. Triangles: $\nu \left(T\right)$ from DKT fits.

Figure 9

Dependence of ${\mu }^{+}$ dynamic relaxation rate $\lambda$ on longitudinal magnetic field $H_L$ from EDSKT fits of Eq. (3) to weak-LF $\mu \mathrm{SR}$ data from ${\mathrm{Pr}}_{2+x}{\mathrm{Ir}}_{2-x}{\mathrm{O}}_{7-\delta }$ at representative temperatures. (a) $T=25$ mK. (b) $T=0.1$ K. (c) $T=0.3$ K. (d) $T=1.0$ K. Lines: fits of power laws $\lambda \left(H_L\right)\propto H_L^a$ to the data for ${\mu }_0{H}_L>10$ mT.