Magnetic dilution and domain selection in the $XY$ pyrochlore antiferromagnet ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$

Below $T_N=1.1$ K, the $XY$ pyrochlore ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ orders into a $k=0$ noncollinear, antiferromagnetic structure referred to as the ${\psi }_2$ state. The magnetic order in ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ is known to obey conventional three-dimensional (3D) percolation in the presence of magnetic dilution, and in that sense is robust to disorder. Recently, however, two theoretical studies have predicted that the ${\psi }_2$ structure should be unstable to the formation of a related ${\psi }_3$ magnetic structure in the presence of magnetic vacancies. To investigate these theories, we have carried out systematic elastic and inelastic neutron scattering studies of three single crystals of ${\mathrm{Er}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ with $x=0$ (pure), $0.2 \left(10\%\mathrm{Y}\right)$ and 0.4 ($20\% \mathrm{Y}$), where magnetic ${\mathrm{Er}}^{3+}$ is substituted by nonmagnetic ${\mathrm{Y}}^{3+}$. We find that the ${\psi }_2$ ground state of pure ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ is significantly affected by magnetic dilution. The characteristic domain selection associated with the ${\psi }_2$ state, and the corresponding energy gap separating ${\psi }_2$ from ${\psi }_3$, vanish for ${\mathrm{Y}}^{3+}$ substitutions between $10\% \mathrm{Y}$ and $20\% \mathrm{Y}$, far removed from the three-dimensional percolation threshold of $\sim 60\% \mathrm{Y}$. The resulting ground state for ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ with magnetic dilutions from $20\% \mathrm{Y}$ up to the percolation threshold is naturally interpreted as a frozen mosaic of ${\psi }_2$ and ${\psi }_3$ domains.

Figure 1

Magnetic Bragg peak intensity for $Q$ = (220) as a function of field for the pure, $10\% \mathrm{Y}$ and $20\% \mathrm{Y}$ diluted samples of ${\mathrm{Er}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$. The elastic scattering was isolated by integrating from $-0.2$ to 0.2 in the [00L] direction and from $-0.1$ to 0.1 meV in energy transfer. The insets show the integrated intensity of the long-range ordered component of the scattering for each of the three samples. Examples of the fits used to obtain these intensities are shown in Fig. 1 of the Supplemental Material [38]. The dashed line in the inset of (b) indicates the expected behavior for the $10\% \mathrm{Y}$ sample at 1.75 T.

Figure 2

Spin wave spectra along three different directions in the HHL plane: [HH0], [22L], and [HHH], all at $T=0.1$ K. The spectra are shown for each of the three samples in zero magnetic field and with magnetic fields of 0.5, 1, and 3 T applied along the [1-10] direction. The contour plots along the [HH0], [22L], and [HHH] directions are obtained by integrating the 3D data set, $S(\vec{Q},E)$, over $-0.2\le \left[00\text{L}\right]\le 0.2,1.8\le \left[\text{HH}0\right]\le 2.2$ and $0.2\le \left[\text{HH-2H}\right]\le 0.2$, respectively.

Figure 3

Inelastic intensity as a function of energy at the (220) Bragg position for the pure, $10\% \mathrm{Y}$ and $20\% \mathrm{Y}$ diluted samples of ${\mathrm{Er}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$. These plots are obtained by integrating the 3D data sets $S(\vec{Q},E)$ at $T=0.1$ K over $1.6\le \left[\text{HH0}\right]\le 2.4$ and $-0.5\le \left[\text{00L}\right]\le 0.5$.

Figure 4

(a) The schematic temperature-dilution phase diagram for ${\mathrm{Er}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$. A dilution between $10\% \mathrm{Y}$ and $20\% \mathrm{Y}$ marks a crossover from a pure ${\psi }_2$ state into a frozen mosaic of ${\psi }_2$ and ${\psi }_3$ domains. The six discrete domains allowed by (b) ${\psi }_2$ and (c) ${\psi }_3$ within the $XY$ plane. (d) Schematic illustration of the state at moderate dilution showing the mosaic of ${\psi }_2$ and ${\psi }_3$ domains.