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 ${\mathrm{Dy}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ and ${\mathrm{Ho}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$. 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 ${\mathrm{Dy}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ versus ${\mathrm{Ho}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$, 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 ${\mathrm{Dy}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ a significant discrepancy between the experimental and Monte Carlo specific-heat results. We discuss possible reasons for this disagreement.

Figure 1

Experimental residual entropy as a function of dilution level $x$ for the ${\mathrm{Dy}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ and ${\mathrm{Ho}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ compounds (adapted from Ke et al. [27]) . Dy denotes ${\mathrm{Dy}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$, Ho denotes ${\mathrm{Ho}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$, and Gen. Pauling denotes the generalized Pauling approximation (gPa) presented in Ref. [27]. As noted by Ke et al., there is an obvious departure between the three curves, except for the undiluted compounds ($x=0$).

Figure 2

Dilution $x$, dependence of the peak height $C_{\mathrm{m}}^{\mathrm{peak}}\left(x\right)$, and peak temperature $T^{\mathrm{peak}}\left(x\right)$ of the magnetic specific heat $C_{\mathrm{m}}\left(T\right)$ of ${\mathrm{Dy}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$. The closed symbols show experimental data while the open symbols are from Monte Carlo simulations of a site-diluted dipolar spin ice model (see text). The circles show the $x$ dependence of $C_{\mathrm{m}}^{\mathrm{peak}}\left(x\right)$ (right axis) while the squares show $x$ dependence of $T^{\mathrm{peak}}\left(x\right)$ on the left axis. A nonmonotonic dependence of $C_{\mathrm{m}}^{\mathrm{peak}}\left(x\right)$ and $T^{\mathrm{peak}}\left(x\right)$ as a function of $x$ is clearly seen in both the experimental and Monte Carlo data. The good agreement between experimental and Monte Carlo data for both quantities that we establish in this work shows that the hypothesis of a simple site-diluted dipolar spin ice model (DSIM) describing ${\mathrm{Dy}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ is reasonably well confirmed up to $x=1.8$. This is one of the main conclusions of our work.

Figure 3

Comparison of the magnetic specific heat $C_{\mathrm{m}}\left(T\right)$ between Monte Carlo simulations and experiments. Black open circles are for ${\mathrm{Dy}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ experiment, solid black curves are for ${\mathrm{Dy}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ simulations. Red open squares are for ${\mathrm{Ho}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ experiment, and solid red curves are for ${\mathrm{Ho}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ simulations. Insets show an enlargement around the Schottky peak at $T_{\mathrm{p}}$, arising from the formation of the spin ice state. The horizontal blue arrows indicate location of $C_{\mathrm{m}}\left(T\right)$ minima that may be occurring in ${\mathrm{Ho}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$.

Figure 4

Residual entropy determined from Monte Carlo simulations for both ${\mathrm{Dy}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ and ${\mathrm{Ho}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ with different low-temperature limits $T_0$. The dotted black curve shows $S_{\mathrm{res}}$ given by the generalized Pauling's argument (gPa).

Figure 5

Comparison of Monte Carlo specific heat with experimental results for ${\mathrm{Dy}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ for sizes $L=3,4,5$ for $x=1.8$ (top panel) and $x=1.9$ (bottom panel). For these simulations, 1000 disorder realizations were considered to carry out the disorder average in Eq. (3).