Nonmonotonic Zero-Point Entropy in Diluted Spin Ice

Water ice and spin ice are important model systems in which theory can directly account for “zero-point” entropy associated with quenched configurational disorder. Spin ice differs from water ice in the important respect that its fundamental constituents, the spins of the magnetic ions, can be removed through replacement with nonmagnetic ions while keeping the lattice structure intact. In order to investigate the interplay of frustrated interactions and quenched disorder, we have performed systematic heat capacity measurements on spin ice materials which have been thus diluted up to 90%. Investigations of both Ho and Dy spin ices reveal that the zero-point entropy depends nonmonotonically on dilution and approaches the value of $R\ln 2$ in the limit of high dilution. The data are in good agreement with a generalization of Pauling’s theory for the entropy of ice.

Figure 1

The real part of ac susceptibility, ${\chi }^{\prime }$, as a function of temperature for diluted ${\mathrm{Dy}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ samples with $x=0$, 0.4, 1.4, 1.8 at a characteristic frequency of 1 kHz and zero field.

Figure 2

Magnetic specific heat $C_{\mathrm{mag}}$ as a function of temperature for diluted ${\mathrm{Dy}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ samples with $x=0$, 0.4, 1.4, 1.8 at $H=0\mathrm{T}$ (a) and $H=1\mathrm{T}$ (b). The inset shows the low temperature dependence of the real part of the $f=50\mathrm{Hz}$ ac susceptibility, $\chi$, measured in a dilution fridge.

Figure 3

The integrated magnetic entropy as a function of temperature for diluted ${\mathrm{Dy}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ samples with $x=0$, 0.4, 1.4, 1.8 at $H=0\mathrm{T}$ (a) and $H=1\mathrm{T}$ (b).

Figure 4

The entropy at $T=16\mathrm{K}$ as a function of dilution $x$ for ${\mathrm{Dy}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ and ${\mathrm{Ho}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ samples at $H=0\mathrm{T}$. The dashed purple curve represents the theoretical calculation based on the Pauling estimate.