Structure and magnetism in the bond-frustrated spinel $\mathrm{ZnC}{\mathrm{r}}_2\mathrm{S}{\mathrm{e}}_4$

The crystal and magnetic structures of stoichiometric $\mathrm{ZnC}{\mathrm{r}}_2\mathrm{S}{\mathrm{e}}_4$ have been investigated using synchrotron x-ray and neutron powder diffraction, muon spin relaxation $\left(\mu \mathrm{SR}\right)$, and inelastic neutron scattering. Synchrotron x-ray diffraction shows a spin-lattice distortion from the cubic $Fd\overline{3}m$ spinel to a tetragonal $I{4}_1/amd$ lattice below $T_N=21\mathrm{K}$, where powder neutron diffraction confirms the formation of a helical magnetic structure with magnetic moment of $3.04\left(3\right){\mu }_{\mathrm{B}}$ at 1.5 K, close to that expected for high-spin $\mathrm{C}{\mathrm{r}}^{3+}$. $\mu \mathrm{SR}$ measurements show prominent local spin correlations that are established at temperatures considerably higher $(<100\mathrm{K})$ than the onset of long-range magnetic order. The stretched exponential nature of the relaxation in the local spin-correlation regime suggests a wide distribution of depolarizing fields. Below $T_N$, unusually fast $(>100\mu {\mathrm{s}}^{-1})$ muon relaxation rates are suggestive of rapid site hopping of the muons in static field. Inelastic neutron scattering measurements show a gapless mode at an incommensurate propagation vector of $k=\left[000.4648\right(2\left)\right]$ in the low-temperature magnetic ordered phase that extends to 0.8 meV. The dispersion is modeled by a two-parameter Hamiltonian, containing ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor interactions with a $J_{nnn}/J_{nn}=-0.337$.

Figure 1

Synchrotron x-ray powder diffraction of $\mathrm{ZnC}{\mathrm{r}}_2\mathrm{S}{\mathrm{e}}_4$ showing (a) the observed, calculated, and difference plot of the Rietveld refinement at 5 K and a comparison of a section of the pattern at (b) 50 K and (c) 5 K highlighting the tetragonal distortion to an $I{4}_1/amd$ space group at low temperature.

Figure 2

Structural parameters obtained from fitting of the synchrotron powder x-ray diffraction data showing that (a) both the Zn-Se and Cr-Se bond lengths are insensitive to changes with temperature, whereas there is a significant modification of their bond angles through the magnetoelastic transition at $T_N=21\mathrm{K}$ and (b) the changes in the lattice parameters at the $Fd\overline{3}m\to I{4}_1/amd$ transition are accompanied by an increase in the Cr thermal factors.

Figure 3

(a) Observed, calculated, and difference plot of the powder neutron diffraction pattern of $\mathrm{ZnC}{\mathrm{r}}_2\mathrm{S}{\mathrm{e}}_4$ at 5 K at $\lambda =2.078\AA$ using the BT1 diffractometer. The main magnetic reflections are marked with an asterisk. Inset highlights the low angle region, indexed in the tetragonal cell. (b) Observed, calculated, and difference plot of the powder neutron diffraction pattern of $\mathrm{ZnC}{\mathrm{r}}_2\mathrm{S}{\mathrm{e}}_4$ at 1.5 K using −0.2 to 0.2-meV elastic contribution of the DCS at $\lambda =5.0\AA$. Inset to (b) shows a graphical representation of the Cr moments within the spinel structure showing two-dimensional planes of ferromagnetic order that propagate in a helical fashion down the $c$ axis.

Figure 4

(a) Early relaxation times for ZF-$\mu \mathrm{SR}$ of $\mathrm{ZnC}{\mathrm{r}}_2\mathrm{S}{\mathrm{e}}_4$ showing very fast relaxation and small initial dip, and (b) example data of 0.3-T LF-$\mu \mathrm{SR}$.

Figure 5

Zero-field measurement results of relaxation rates of the two components below $T_N$ and the value of the relaxation rate and power factor of the stretched exponent above $T_N$. Inset shows the temperature dependence of the fast relaxing component and the oscillation frequency.

Figure 6

Fit results for the 0.3-T (3-kG) longitudinal field measurement. At all temperatures two fractions are present with amplitudes ${\lambda }_1$ (fast) and ${\lambda }_2$ (slow). Above $T_N$ (from the region marked by the circle) they can be replaced by one stretched exponential form of relaxation.

Figure 7

Temperature dependence of the powder-averaged magnetic fluctuations taken with $E_i=1.7\mathrm{meV}$ on DCS. Panel (a) illustrates the spectrum in the magnetic ordered state at 2 K, with (b) showing that these are replaced by momentum and energy-broadened excitations above $T_N$. (c) High-temperature scan illustrating the background.

Figure 8

(a) $E_i=1.33\mathrm{meV}$ data from DCS with the solid line and open points being the predicted single-crystal dispersion along $c$ and within the a-b plane. (b) Lower-resolution $E_i=3.3\mathrm{meV}$ data illustrating a high-energy branch extending up to 1.5 meV. (c) Plot of the peak position as a function of $Q^2$ illustrating a fit to a ferromagnetic dispersion curve $(\propto Q^2)$.

Figure 9

(a) Fit with numerous models to the evolution of magnetic moment in $\mathrm{ZnC}{\mathrm{r}}_2\mathrm{S}{\mathrm{e}}_4$ as a function of temperature, showing an unusual exponent of $\beta =0.14\left(1\right)$, where the development of magnetic ordering occurs much quicker than expected within the Heisenberg model as a result of the short-ranged ordering occurring below 100 K. (b) Evolution of the lattice and structural parameters in $\mathrm{ZnC}{\mathrm{r}}_2\mathrm{S}{\mathrm{e}}_4$ as a function of temperature.

Figure 10

Reconstruction of the electron density of $\mathrm{ZnC}{\mathrm{r}}_2\mathrm{S}{\mathrm{e}}_4$ from ID31 synchrotron powder x-ray diffraction data at 4 K using the maximum entropy method. No assumptions of symmetry are made during this analysis and the isotropic densities of all of the atoms present are indicative of little significant site disorder. Rietveld refinement of the neutron and synchrotron data shows the materials were stoichiometric as made.