Nonmonotonic residual entropy in diluted spin ice: A comparison between Monte Carlo simulations of diluted dipolar spin ice models and experimental results

Spin ice materials, such as ${\mathrm{Dy}}_{2}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}$ and ${\mathrm{Ho}}_{2}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}$, are highly frustrated magnetic systems. Their low-temperature strongly correlated state can be mapped onto the proton disordered state of common water ice. As a result, spin ices display the same low-temperature residual Pauling entropy as water ice, at least in calorimetric experiments that are equilibrated over moderately long-time scales. It was found in a previous study [X. Ke et al., Phys. Rev. Lett. 99, 137203 (2007)] that, upon dilution of the magnetic rare-earth ions (${\mathrm{Dy}}^{3+}$ and ${\mathrm{Ho}}^{3+}$) by nonmagnetic yttrium (${\mathrm{Y}}^{3+}$) ions, the residual entropy depends nonmonotonically on the concentration of ${\mathrm{Y}}^{3+}$ ions. A quantitative description of the magnetic specific heat of site-diluted spin ice materials can be viewed as a further test aimed at validating the microscopic Hamiltonian description of these systems. In this work, we report results from Monte Carlo simulations of site-diluted microscopic dipolar spin ice models (DSIM) that account quantitatively for the experimental specific-heat measurements, and thus also for the residual entropy, as a function of dilution, for both ${\mathrm{Dy}}_{2\ensuremath{-}x}{\mathrm{Y}}_{x}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}$ and ${\mathrm{Ho}}_{2\ensuremath{-}x}{\mathrm{Y}}_{x}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}$. The main features of the dilution physics displayed by the magnetic specific-heat data are quantitatively captured by the diluted DSIM up to 85% of the magnetic ions diluted ($x=1.7$). The previously reported departures in the residual entropy between ${\mathrm{Dy}}_{2\ensuremath{-}x}{\mathrm{Y}}_{x}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}$ versus ${\mathrm{Ho}}_{2\ensuremath{-}x}{\mathrm{Y}}_{x}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}$, as well as with a site-dilution variant of Pauling's approximation, are thus rationalized through the site-diluted DSIM. We find for 90% ($x=1.8$) and 95% ($x=1.9$) of the magnetic ions diluted in ${\mathrm{Dy}}_{2\ensuremath{-}x}{\mathrm{Y}}_{x}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}$ a significant discrepancy between the experimental and Monte Carlo specific-heat results. We discuss possible reasons for this disagreement.

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