Second-order phase transition in FeCr2S4investigated by Mössbauer spectroscopy: An example of orbital para-to-ferromagnetism transition

Mössbauer studies on stoichiometric samples of the spinel ferrimagnet Fe${\mathrm{Cr}}_2$ ${\mathrm{S}}_4$ are reported with special attention to the second-order phase transition at 13 K. Magnetic spectra were first solved as a superposition of singlets. This gives the experimental linewidths and intensities of each line. These as-deduced linewidths and intensities were assumed in computing the magnetic spectra for a single-site solution. The transition at 13 K is associated to a discontinuity of the barycenter of the spectra. Moreover, anomalies of the linewidth variations versus $T$ as well as differences between the theoretical and the experimental intensities of the peaks cannot be resolved in the single-site solution. The singleness of this solution is discussed. A more reliable two-site solution is suggested. The experimental mean linewidth, the characteristics of the electric-field-gradient (EFG) ($V_{\mathrm{zz}}$, $\eta$, and $\theta$), and the hyperfine magnetic field of the sites show discontinuities at 13 K. These features are explained assuming narrow $d$ bands in Fe${\mathrm{Cr}}_2$ ${\mathrm{S}}_4$. The sixth $3d$ electron of $A$-site ${\mathrm{Fe}}^{2+}$ is assumed to occupy an orbital with $|\epsilon 〉$ and $|\theta 〉+|{\psi }_{{\mathrm{Cr}}^{2+}}〉$ symmetry due to the hybridization of $|\theta 〉$ with the $B$-site ${\mathrm{Cr}}^{2+}$ state. ${\mathrm{Fe}}_A^{2+}$ and ${\mathrm{Cr}}_B^{2+}$ in thiospinel lie almost at the same energy level. At $T>\mathrm{}13\mathrm{}$ K, the sixth $3d$ electron shares the $|\epsilon 〉$ and the hybridized $|\theta 〉$ orbital statistically. This gives two sites with the same intensity. At $T<\mathrm{}13\mathrm{}$ K, only the hybridized $|\theta 〉$ orbital is occupied. This corresponds to an orbital paramagnetism ($T>\mathrm{}13\mathrm{}$ K) to a ferromagnetic ordering ($T<\mathrm{}13\mathrm{}$ K) transition. Recently Cyrot showed that an Hubbard Hamiltonian in the atomic limit leads not only to spin ordering but also to orbital ordering. Furthermore, the transition at $T<\mathrm{}13\mathrm{}$ K is associated to a slow relaxation of the EFG. The coupling of the band with the phonons at $T<\mathrm{}13\mathrm{}$ K is different from a linear and weak dynamic Jahn-Teller effect as is the case for $A$-site diluted ${\mathrm{Fe}}^{2+}$.