Spin dynamics in the ordered spin ice Tb${}_2$Sn${}_2$O${}_7$

Inelastic neutron scattering experiments have been carried out in the “ordered spin ice” Tb${}_2$Sn${}_2$O${}_7$, aiming at a detailed understanding of the low-energy magnetic excitations in this highly frustrated magnet. Besides the crystal-field transition lines, we report on a well-defined low-energy line that develops in the ordered phase. We show that the intensity of this transition cannot be understood within the trigonal crystal-field scheme and argue that an interaction, which involves the spin lattice coupling through a low-temperature distortion of the trigonal crystal field, is necessary to explain the data. Based on random-phase approximation spin-dynamics calculations, taking into account the actual crystal-field scheme as well as this distortion, we successfully describe the neutron data, quantitatively consistent with the experimental observations.

Figure 1

In the “soft spin ice” model (Ref. 14), the evolution of the magnetic structure as the anisotropy is decreased can be monitored by the variation of the moment angles with respect to their local anisotropy axes, or equivalently by the evolution of the normalized magnetization $r=M_0/M_{\mathrm{sat}}$ along $\langle 001\rangle$. The ratio $r$ varies from $1/\sqrt{3}=0.577$ to 1 when going from Ising to Heisenberg behavior. (a) “Two-in–two-out” configuration typical of the canonical “spin ice” ground state with magnetic moments along the local $\langle 111\rangle$ directions. (b) Standard ordered ferromagnetic configuration. (c) “Ordered spin ice” ground state as found in Tb${}_2$Sn${}_2$O${}_7$.

Figure 2

(Left) Map of the powder average $\mathcal{S}(Q,\omega )$ in Tb${}_2$Sn${}_2$O${}_7$ measured at 1.3 K (top) and 65 mK (middle) and a three-dimensional map at 65 mK (bottom). Data were taken on the IN5 spectrometer at ILL with an incident wavelength $\lambda =5$ Å. (Right) Calculated powder average $\mathcal{S}(Q,\omega )$ with a tetragonal distortion of the trigonal CEF ($D_Q=0.2$ K) and an antiferromagnetic exchange constant $\mathcal{J}=-0.09$ K, in the paramagnetic phase ($T=5$ K, top) and in the OSI phase ($T=65$ mK, middle). The calculation at $T=65$ mK with $D_Q=0$ (bottom) shows that, without distortion, the low-energy line at 0.35 meV has a vanishingly small intensity, contrary to experiment.

Figure 3

(a) Energy cuts for a wave vector transfer $Q=0.5$ Å${}^{-1}$ and several temperatures in the OSI phase. These cuts are obtained from the map by integrating over a $\pm 0.04$ Å${}^{-1}$ wide band around $Q$. (b) Temperature evolution of the peak position around 0.4 meV along with its energy width, deduced from phenomenological Lorentzian fits performed at several temperatures.

Figure 4

Experimental (left) and calculated (right) inelastic spectra at 65 mK, for $Q=0.5$ and $1.5$ Å${}^{-1}$, in Tb${}_2$Sn${}_2$O${}_7$. The simulations were performed with $\mathcal{J}=-0.09$ K and $D_Q=0.2$ K. A constant background was added to the calculated profiles, and their overall intensity was arbitrarily scaled to reproduce the intensity of the upper band at $Q=1$ Å${}^{-1}$.

Figure 5

Ordered spin ice part of the ($\mathcal{J},T_{\mathrm{C}}$) phase diagram computed using the crystal-field interaction in Tb${}_2$Sn${}_2$O${}_7$ in the presence of the tetragonal distortion (8), for different values of the $D_Q$ parameter. The upper abscissa scale is $J_{nn}/D_{nn}$, as defined in Ref. 28. The link with the exchange constant $\mathcal{J}$ is $J_{nn}/D_{nn}=\frac{4\pi }{5{\mu }_0}\frac{\mathcal{J}{k}_B{d}^3}{g_J^2{\mu }_B^2}$, where $d$ is the nearest-neighbor distance; that is, $J_{nn}/D_{nn}=6.60\mathcal{J}$ (K).

Figure 6

Calculated neutron scattering intensity with the trigonal CEF of Tb${}_2$Sn${}_2$O${}_7$ for different orientations of the molecular field $\vec{h}$ with respect to the local $\langle$$111\rangle$ axis. The calculation does not reproduce the spectral weight of the experimental observed excitation at 0.35 meV, marked by a magenta arrow.