Magnetic Density of States at Low Energy in Geometrically Frustrated Systems

Using muon-spin-relaxation measurements we show that the pyrochlore compound ${\mathrm{Gd}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$, in its magnetically ordered phase below $\sim 1\mathrm{K}$, displays persistent spin dynamics down to temperatures as low as 20 mK. The characteristics of the induced muon relaxation can be accounted for by a scattering process involving two magnetic excitations, with a density of states characterized by an upturn at low energy and a small gap depending linearly on the temperature. We propose that such a density of states is a generic feature of geometrically frustrated magnetic materials.

Figure 1

$\mu \mathrm{SR}$ spectrum recorded in zero field at 20 mK in a single crystal of ${\mathrm{Gd}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ with the initial muon beam polarization parallel to the $[111]$ crystal direction. The main frame displays the time spectrum up to $8\mu \mathrm{s}$, and the inset magnifies the short time behavior. The latter is characterized by wiggles which are damped out at the earliest time displayed in the main frame. The wiggles correspond to the spontaneous precession of the muon spin in two local fields of respective magnitude $B_1=194(1)$ and $B_2=149(1)\mathrm{mT}$. As expected, the spin-lattice relaxation channel, most easily seen in the main frame, accounts for $\sim 1/3$ of the spectrum and the wiggles for $\sim 2/3$, this being temperature independent as is also the total asymmetry. The amplitude of the $B_1$ component is approximately 4 times as large as that of $B_2$.

Figure 2

In ${\mathrm{Gd}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$, low temperature dependence of specific heat over temperature $C_p/T$ per mole of Gd (top panel), of the muon-spin-lattice relaxation rate ${\lambda }_Z$ (middle panel), and of the exponent $\alpha$ (bottom panel) of the stretched exponential decay. The muon data were measured in a single crystal, with different orientations relative to the muon polarization ${\mathbf{S}}_{\mu }$, as shown, and the anisotropy of ${\lambda }_Z$ was found to be negligible. The dashed lines for ${\lambda }_Z(T)$ and $\alpha (T)$ are guides for the eye. The lower inset shows, on a double logarithmic scale, the weak temperature dependence of ${\lambda }_Z(T)$ at low temperature. The full lines give the predictions for $C_p/T$ and ${\lambda }_Z$ computed with $b_{\mu }=0.03{\mathrm{meV}}^{-1/2}$, $b_{\mathrm{sh}}=26{\mathrm{meV}}^{-2}$ per Gd atom and $a=0.1$ (see text). The upper inset shows the density of magnetic excitations $g_m(ϵ)$ at 50 mK, with its two components as dashed lines. The gap is: $\Delta \simeq 5\times {10}^{-4}\mathrm{meV}$. The upturn at low energy is responsible for the finite value of ${\lambda }_Z$ as $T\to 0$.