Magnetic excitations in the spinel compound ${\text{Li}}_x\left[{\text{Mn}}_{1.96}{\text{Li}}_{0.04}\right]{\text{O}}_4$ $\left(x=0.2,0.6,0.8,1.0\right)$: How a classical system can mimic quantum critical scaling

We present neutron-scattering results on the magnetic excitations in the spinel compounds ${\text{Li}}_x\left[{\text{Mn}}_{1.96}{\text{Li}}_{0.04}\right]{\text{O}}_4$ $\left(x=0.2,0.6,0.8,1.0\right)$. We show that the dominant excitations below $T\sim 70\text{K}$ are determined by Mn ions located in clusters, and that these excitations mimic the dynamic scaling found in quantum critical systems that also harbor magnetic clusters, such as ${\text{CeRu}}_{0.5}{\text{Fe}}_{1.5}{\text{Ge}}_2$. We argue that our results for this classical spinel compound suggest that the unusual response at low temperatures as observed in quantum critical systems that have been driven to criticality through substantial chemical doping is (at least) partially the result of the fragmentation of the magnetic lattice into smaller units.

Figure 1

(Color online) Figure reproduced from Ref. 18. The magnetic phase diagram in the vicinity of a QCP (black dot) as a function of the coupling strength $J$ between local moments and the electron spins. This coupling strength can be tweaked by applying hydrostatic or chemical pressure. The striped areas indicate that the moments are fully shielded, the solid areas are the region where long-range order is present. (a) The Hertz-Millis scenario (Refs. 10, 11) where ordering is established through a spin-density wave instability of the Fermi surface. (b) The local moment scenario (Refs. 12, 17) where the QCP is the point where moments can first survive, with decreasing $J$, all the way down to 0 K. (c) Local variations in interatomic distances have resulted in a spread in shielding temperatures yielding a region of the phase diagram between where moments are completely shielded and where the (partially shielded) moments order (Ref. 18). In this region the surviving (down to 0 K) moments are located in clusters that are not interconnected, as shown in part (e). In this latter scenario, the QCP would correspond to the presence of a lattice spanning cluster, as depicted in part (d).

Figure 2

(Color online) (a) The unit cell for stoichiometric ${\text{LiMn}}_2{\text{O}}_4$ projected along the $c$ axis. The ${\text{Mn}}^{4+}$-ions (open circles) form isolated eightfold rings, Ref. 31, while the ${\text{Mn}}^{3+}$ ions (filled circles) form columns along the $c$ direction. The numbers within the rings denote their $z$ position in the unit cell (b) Two unit cells stacked along the $c$ axis. Two isolated ring clusters are shown at the top of the figure. Upon Li substitution on the Mn sites (or Mn removal), some ${\text{Mn}}^{3+}$ ions will become ${\text{Mn}}^{4+}$ ions, leading to modified rings (Ref. 29) and linked rings, two possibilities of which are shown in the bottom half of the figure. (c) The observed elastic scattering in $\text{Li}\left[{\text{Mn}}_{1.96}{\text{Li}}_{0.04}\right]{\text{O}}_4$ at 30 K (Ref. 38) (circles). The two-ring clusters [linked along the $c$ direction; solid line, corresponding to the bottom left cluster in Fig. 2b or linked along the $ab$ direction; dashed line, corresponding to the bottom right cluster in Fig. 2b] give a satisfactory description (Ref. 29) of the neutron data.

Figure 3

Details of representative data sets for the new ILL experiments showing the scattering for $x=1.0$ in the window $60°<\varphi <80°$ at 150 K (filled circles) at 41 K (open circles), the empty container scattering (dashed-dotted curve) and the time-independent background (solid line) as obtained from an empty spectrometer run. As can be seen, the container scattering is negligible for $\left|E\right|>0.5$, and the data correction only requires subtraction of the time-independent background for $\left|E\right|>0.5$, the range we used to test dynamic scaling laws.

Figure 4

(Color online) The TRIAX data for $x=1.0$ after background correction for temperatures indicated in the figure. The critical scattering associated with cluster formation around $\sim 70\text{K}$ (Refs. 29, 32) shows up as increased intensity in the inelastic channels. Note that this critical scattering extends over a wide range of temperatures, which in turn reflects the frustration (Ref. 27) inherent in systems of this type.

Figure 5

(Color online) The magnetic form factor (circles) is obtained as the energy integrated scattering, $M\left(q\right)={\int }_{-0.15\text{meV}}^{0.15\text{meV}}I\left(q,E,T=24\text{K}\right)dE$, associated with the magnetic clusters for four compositions given in the figure. This scattering, determined from the total magnetic scattering at $T=24\text{K}$ in the energy window $\left|E\right|<0.15\text{meV}$ represents the slowest—or even frozen in—dynamics of the clusters, yielding the magnetic form factor of the clusters. The almost completely lithium depleted sample $\left(x=0.2\right)$ displays long-range order [sharp Bragg peaks (Ref. 37)], as well as a substantial amount of short-range order. The data have been normalized to sample weight, but an absolute normalization was not possible. For reference, the absolutely normalized data at $x=1.0$ (scaled to give the best agreement with the new ILL data) obtained from the pilot experiment (Ref. 38) are shown by the solid lines in each figure.

Figure 6

(Color online) The magnetic form factor $M\left(q\right)$ for the $x=1.0$ sample at 50 K as determined from energy integration over the energy transfers shown in the plot. In order to evaluate the shape of the resultant curves, we have normalized each curve to the average intensity in the $q$ range $1.25<q<1.5{\text{Å}}^{-1}$.

Figure 7

(Color online) (a) The energy dependence of the scattered intensity $I\left(q,E\right)$ for 13 $q$ values $0.5<q<1.8{\text{Å}}^{-1}$ at $T=30\text{K}$ for the pilot experiment (Ref. 38). The energy resolution is given by the sharp incoherent nuclear peak. The data have been normalized to the magnetic form factor $M\left(q\right)$ of the clusters, which has been determined from $M\left(q\right)=I\left(q,E=0.6\text{meV}\right)$ (see text). All data lie on a single curve, implying that the cluster dynamics are independent of $q$. (b)–(f) same as part (a), but now for the new ILL data taken at $x=1.0$, 0.8, 0.6, and 0.2, respectively, at $T=33\text{K}$ [part (b)] and at $T=41\text{K}$ [parts (c) through (f)]. The solid lines in part (b)–(f) are the smoothed data from part (a) to show the consistency between the data sets at $T\sim 30\text{K}$ and to provide a reference for $T=41\text{K}$. The scatter of the data points in part (c) is because the particular experimental run was shorter than the ones shown in the other parts of the figure.

Figure 8

(Color online) Same as Fig. 7, but now the data have been corrected for (trivial) temperature effects [Eq. (2)] and they have been plotted versus $E/k_{B}T$ (circles). The data in part (a) are the ILL data of the pilot experiment, Ref. 38, parts (b)–(e) correspond to the new ILL experiments. The Li concentrations $x$ and temperatures are shown in the figure. All curves collapse into one curve, giving the appearance of dynamical scaling. The finite energy resolution (Fig. 3) of the spectrometer causes the apparent differences between the temperatures for $\left|E/k_{B}T\right|<0.1$.

Figure 9

(Color online) (a) The magnetic scattering of the fully lithiated compound $\left(x=1.0\right)$ measured at $q=1.4{\text{Å}}^{-1}$ on TRIAX. The intensity scale is the same as for in Fig. 4. The scattering behaves as if $E/T$ is the only relevant variable: the scattering intensity develops along straight lines $E\sim T$ such as the white line $E=k_{B}T$ shown in the figure. (b) When $g\left(E/T\right)$ [Eq. (3)] is plotted on a log-log scale, the scattering appears to display dynamical scaling over the full range. The lower curve are the ILL data from Fig. 8a, the middle curve are all the TRIAX data shown in part (a) of this figure, and the upper curve are the data for $\text{Ce}{\left({\text{Ru}}_{0.24}{\text{Fe}}_{0.76}\right)}_2{\text{Ge}}_2$ at $q=0.4{\text{Å}}^{-1}$ (Ref. 15). The three curves are offset for plotting clarity.