Pauling Entropy, Metastability, and Equilibrium in ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ Spin Ice

Determining the fate of the Pauling entropy in the classical spin ice material ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ with respect to the third law of thermodynamics has become an important test case for understanding the existence and stability of ice-rule states in general. The standard model of spin ice—the dipolar spin ice model—predicts an ordering transition at $T\approx 0.15\mathrm{K}$, but recent experiments by Pomaranski et al. suggest an entropy recovery over long timescales at temperatures as high as 0.5 K, much too high to be compatible with the theory. Using neutron scattering and specific heat measurements at low temperatures and with long timescales ($0.35\mathrm{K}/{10}^6\mathrm{s}$ and $0.5\mathrm{K}/{10}^5\mathrm{s}$, respectively) on several isotopically enriched samples, we find no evidence of a reduction of ice-rule correlations or spin entropy. High-resolution simulations of the neutron structure factor show that the spin correlations remain well described by the dipolar spin ice model at all temperatures. Furthermore, by careful consideration of hyperfine contributions, we conclude that the original entropy measurements of Ramirez et al. are, after all, essentially correct: The short-time relaxation method used in that study gives a reasonably accurate estimate of the equilibrium spin ice entropy due to a cancellation of contributions.

Figure 1

Measurement of spin correlations with accurately equilibrated spin temperature of 0.65 K. (a) Example in situ susceptibility measurements (susceptibility $\chi$ is a linear function of the voltage). (b) The measured (lower hemisphere) and simulated (upper hemisphere) neutron structure factors are in close agreement. (c) Comparison of measured and calculated intensities (the color scale indicates the point density, with 11 518 points in total).

Figure 2

Slow cooling and equilibration at 0.35 K. (a) The sample was cooled by a protocol similar to Ref. [20]. (b) Comparing a rapidly cooled measurement at 0.35 K with the slow-cooled measurement by taking the difference of the structure factors shows no difference between the two [data are normalized to match the scale of Fig. 1]. (c) This is highlighted by comparing cuts across (0,0,3) in the quenched and difference maps.

Figure 3

Specific heat of ${}^{162}\mathrm{Dy}$, ${}^{\mathrm{Nat}}\mathrm{Dy}$, and ${}^{163}\mathrm{Dy}$ samples of ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. (a) Long- and short-time measurements. (b) Difference between long- and short-time measurements and calculated nuclear specific heat for the ${}^{\mathrm{Nat}}\mathrm{Dy}$ sample. (c) Equilibrium electronic specific heat for all samples, as well as model calculation and previous results from Ref. [20] (adjusted for the correct hyperfine contribution [19]). In (b) and (c), the lowest temperature displayed, 0.48 K, is the low-temperature limit of our long-time measurements.