Gapless quantum excitations from an icelike splayed ferromagnetic ground state in stoichiometric ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$

The ground state of the quantum spin ice candidate magnet ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ is known to be sensitive to weak disorder at the $\sim 1\%$ level which occurs in single crystals grown from the melt. Powders produced by solid state synthesis tend to be stoichiometric and display large and sharp heat capacity anomalies at relatively high temperatures, $T_C\sim 0.26$ K. We have carried out neutron elastic and inelastic measurements on well characterized and equilibrated stoichiometric powder samples of ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ which show resolution-limited Bragg peaks to appear at low temperatures, but whose onset correlates with temperatures much higher than $T_C$. The corresponding magnetic structure is best described as an icelike splayed ferromagnet. The spin dynamics in ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ are shown to be gapless on an energy scale $<0.09$ meV at all temperatures and organized into a continuum of scattering with vestiges of highly overdamped ferromagnetic spin waves present. These excitations differ greatly from conventional spin waves predicted for ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$'s mean field ordered state, but appear robust to weak disorder as they are largely consistent with those displayed by nonstoichiometric crushed single crystals and single crystals, as well as by powder samples of ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$'s sister quantum magnet ${\mathrm{Yb}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$.

Figure 1

The temperature dependence of the (111) and (113) Bragg peaks. Panels (a) and (c) show the elastically scattered neutron intensity as a function of the momentum transfer $|\vec{Q}|$ around the (111) and (113) Bragg peaks respectively. Panels (b) and (d) shows the subtraction between the different $|\vec{Q}|$ cuts at different temperatures. A clear, $|\vec{Q}|$-resolution-limited increase of the elastic scattering at these two Bragg peaks is observed and is representative of the data used to model the magnetic structure of stoichiometric ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. Error bars represent one standard deviation.

Figure 2

Intensity of elastic magnetic scattering vs $Q$ and the corresponding fit to the splayed ice ferromagnet. (a) the best fit of the model for a splayed ice ferromagnet to the difference of elastic neutron scattering intensity at 100 mK and 8 K (i.e., the magnetic elastic scattering) is shown. The model employed is a splayed ferromagnet (a $Q=0$ structure with a net moment along the cubic axes) on the pyrochlore lattice, with splay angle of $\theta =14{\left(5\right)}^{\circ }$ and an ordered moment of 0.90(9) ${\mu }_B$. Note: the error bars are not shown in this figure for clarity, but are taken into account as weights in the least squared fitting of the model. (b) The goodness-of-fit parameter ${\chi }^2$ as a function of splay angle is shown. The inset to (b) shows an illustration of the best fit magnetic structure model on a pair of tetrahedra.

Figure 3

The temperature dependence of the elastic scattered intensity at $|\vec{Q}|=\left(111\right)$. The scattered elastic neutron intensity, relative to $T=700$ mK at $|\vec{Q}|=\left(111\right)$, shows similar temperature dependence for both a previously measured single crystal (data taken from Ross et al. [14]) and for the stoichiometric powder sample of ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ (this paper). With increasing temperature, the elastic magnetic scattering begins to fall around $T=0.35$ K and decreases approximately linearly above this temperature. The specific heat anomaly $T_C\sim 0.26$ K for a representative powder sample of ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ (data taken from Ross et al. [28]) is shown for reference and reveals the relative insensitivity of the temperature dependence of the elastic scattering to the specific heat anomaly.

Figure 4

$S\left(\right|\vec{Q}|,E)$ as measured for stoichiometric and weakly stuffed ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ with comparison to ${\mathrm{Yb}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$. (a) and (b) show $S\left(\right|\vec{Q}|,E)$ measured on the stoichiometric ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ powder and the lightly stuffed CSC powder of ${\mathrm{Yb}}_{2+x}{\mathrm{Ti}}_{2-x}{\mathrm{O}}_{7+y}$ with $x=0.046$, respectively, both at $T=0.1$ K. (c) shows the corresponding $S\left(\right|\vec{Q}|,E)$ measurement on single crystal ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$, also at $T=0.1$ K. Panel (d) shows $S\left(\right|\vec{Q}|,E)$ measured on powder ${\mathrm{Yb}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ at $T=0.1$ K, by Dun et al. [37]. All four data sets have had an empty can background data set subtracted from them and have had intensities scaled such that the scattering at $\overrightarrow{\left|Q\right|}$ = $1{\text{Å}}^{-1}$ and $E=0.5$ meV saturates the color scale.

Figure 5

Energy vs $|\vec{Q}|$ slice of the mean-field calculation of $S\left(\right|\vec{Q}|,E)$ for ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. $S\left(\right|\vec{Q}|,E)$ has been computed using an anisotropic spin 1/2 exchange Hamiltonian with the exchange parameters determined by Ross et al. [12]. This calculation successfully accounts for the spin wave spectrum in the high magnetic field, polarized state at all wave vectors, but does not resemble the inelastic neutron scattering seen in Fig. 4. In particular the measured inelastic magnetic scattering is gapless at all wave vectors, while the calculated $S\left(\right|\vec{Q}|,E)$, shown here, possesses a minimum gap of $\sim 0.25$ meV.

Figure 6

Energy cuts of $S\left(\right|\vec{Q}|,E)$ for the stoichiometric powder sample of ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and the CSC ${\mathrm{Yb}}_{2+x}{\mathrm{Ti}}_{2-x}{\mathrm{O}}_{7+y}$ with $x=0.046$. Top panels: These four different panels show cuts through $S\left(\right|\vec{Q}|,E)$ of ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$, taken from Fig. 4, for different ranges of $|\vec{Q}|$ and at different temperatures. Four different integrations in $|\vec{Q}|$ going from smaller $|\vec{Q}|$ in (a) to larger $|\vec{Q}|$ in (d) have been applied, as indicated in the individual panels. No differences in the scattering are observed from $T=2.5$ K down to $T=0.1$ K. Bottom panels: Same data and $|\vec{Q}|$ integrated cuts as in (a)–(d) but for ${\mathrm{Yb}}_{2+x}{\mathrm{Ti}}_{2-x}{\mathrm{O}}_{7+y}$ with $x=0.046$ taken from Fig. 4.