Dynamics of diluted Ho spin ice ${\mathrm{Ho}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ studied by neutron spin echo spectroscopy and ac susceptibility

We have studied the spin relaxation in diluted spin ice ${\mathrm{Ho}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ by means of neutron spin echo spectroscopy and ac susceptibility measurements. Remarkably, the geometrical frustration is not relieved by doping with nonmagnetic Y, and the dynamics of the freezing is unaltered in the spin echo time window up to $x\simeq 1.6$. At higher doping with nonmagnetic Y $(x\ge 1.6)$ a new relaxation process at relatively high temperature (up to at least $T\simeq 55\mathrm{K}$) appears which is more than 10 times faster than the thermally activated main relaxation process. We find evidence that over the whole range of composition all Ho spins participate in the dynamics. These results are compared to ac susceptibility measurements of the diluted Ho and Dy spin ice systems, which show that at low temperature the Arrhenius behavior is masked by another dynamical process with little temperature dependence. X-ray absorption fine structure (EXAFS) spectra and x-ray diffraction show that the samples are structurally well ordered.

Figure 1

The rare-earth sublattice of corner-sharing tetrahedra in the pyrochlore structure.

Figure 2

(Color online) The normalized intermediate scattering function $S(Q,t)∕S(Q,0)$ for four samples (including the pure spin ice) at different temperatures. Lines are fits to single or double exponential relaxation functions (see text). Note that the time axis in the bottom left-hand panel is different. The NIST experiment reached a correlation time of $2\times {10}^{-8}\mathrm{s}$, 10 times longer than at IN11.

Figure 3

The normalized intermediate scattering function $S(Q,t)∕S(Q,0)$ at two different temperatures for the sample with highest doping $(x=1.8)$ showing the absence of $Q$-dependence in the dynamics.

Figure 4

(Color online) The ac susceptibility of $x=1.8$ $(\nabla )$, $x=1.5$ $(◻)$, and $x=1.3$ $(▵)$ samples, together with data of pure HTO $(엯)$ and DTO $(◇)$ for comparison. Upper panel (a) imaginary part of ac susceptibility normalized to dc susceptibility. Lower panel (b) inverse of the temperature where the main peak in the imaginary part of ac susceptibility occurs. Lines are guides to the eye.

Figure 5

Combination of neutron spin echo and ac susceptibility results. This graph shows the inverse of the temperatures at which maxima in the imaginary part of the susceptibility occur, as a function of the probing frequency, for the $x=1.3$ sample (the others are qualitatively similar, see Fig. 4). The “high-$T$” peak of the susceptibility is dominant, and in the ${10}^3–{10}^4\mathrm{Hz}$ range the slope of the curve corresponds to a gap of $\Delta \sim 100\mathrm{K}$. For the spin echo frequency range the inverse measurement temperature is plotted against the inverse of the spin relaxation time. The inset shows the NSE data alone for better visibility, for the $x=1.8$ $(▵)$, $x=1.6$ $(◇)$, $x=1.15$ $(엯)$, $x=1.1$ $(\nabla )$, $x=0.7$ $(◁)$, and $x=0$ $(◻)$, samples, revealing Arrhenius nature in the temperature range between $40\mathrm{K}$ and $200\mathrm{K}$. The slope of the curve of the NSE data corresponds to $\Delta \sim 270\mathrm{K}$.

Figure 6

(Color online) Fourier transform (FT) magnitudes of the $k^3\chi \left(k\right)$ EXAFS data for some ${\mathrm{Ho}}_{2-x}{\mathrm{Y}}_x{\mathrm{Ti}}_2{\mathrm{O}}_7$ samples at the (a) Ti $K$ edge and (b) the Y $K$ edge. These data were collected at $30\mathrm{K}$. Data are transformed from 2.5 to $16.0{\mathrm{\AA }}^{-1}$ and Gaussian narrowed by $0.3{\mathrm{\AA }}^{-1}$. The fit ranges are from 1.4 to $4.0\mathrm{\AA }$ in the upper panel and from 1.6 to $5.9\mathrm{\AA }$ in the lower panel, respectively.