Magnetoelasticity in $A$Cr${}_2$O${}_4$ spinel oxides ($A=$ Mn, Fe, Co, Ni, and Cu)

Dynamical properties of the lattice structure were studied by optical spectroscopy in $A$Cr${}_2$O${}_4$ chromium spinel oxide magnetic semiconductors over a broad temperature region of $T=10$–335 K. The systematic change of the $A$-site ions ($A=$ Mn, Fe, Co, Ni and Cu) showed that the occupancy of 3$d$ orbitals on the $A$ site has strong impact on the lattice dynamics. For compounds with orbital degeneracy (FeCr${}_2$O${}_4$, NiCr${}_2$O${}_4$, and CuCr${}_2$O${}_4$), clear splitting of infrared-active phonon modes and/or activation of silent vibrational modes have been observed upon the Jahn-Teller transition and at the onset of the subsequent long-range magnetic order. Although MnCr${}_2$O${}_4$ and CoCr${}_2$O${}_4$ show multiferroic and magnetoelectric character, no considerable magnetoelasticity was found in spinel compounds without orbital degeneracy as they closely preserve the high-temperature cubic spinel structure even in their magnetic ground state. Aside from lattice vibrations, intra-atomic 3$d$-3$d$ transitions of the $A$${}^{2+}$ ions were also investigated to determine the crystal field and Racah parameters and the strength of the spin-orbit coupling.

Figure 1

(a) In the structure of $A$Cr${}_2$O${}_4$ spinels, the Cr${}^{3+}$ ions (light blue spheres) and the $A$${}^{2+}$ ions (gray spheres) are located on a pyrochlore and a diamond lattice, respectively. The oxygens (red spheres) form octahedral environment around Cr${}^{3+}$ ions, while $A$${}^{2+}$ ions are surrounded by tetrahedral oxygen cages. (b) Electron configuration of Cr${}^{3+}$ ion and $A$-site ions with increasing electron number on the 3$d$ shell. Here, the splitting due to the local crystal field of the ions in the cubic phase and the effect of Hund coupling are only taken into account.

Figure 2

Reflectivity spectra (left panel) and the real part of the optical conductivity spectra (right panel) over the photon energy range of 100 cm${}^{-1}$ and 24 500 cm${}^{-1}$ for the $A$Cr${}_2$O${}_4$ ($A=$ Mn, Fe, Co, Ni, and Cu) spinel crystals measured at $T=10$ K. This spectral range covers the in-gap excitations, namely, the optical phonon modes and the intra-atomic 3$d$-3$d$ transitions. The 3$d$-3$d$ transitions of $A$${}^{2+}$ ions are indicated by black triangles, while the 3$d$-3$d$ transitions of Cr${}^{3+}$ ions are shown with orange triangles. The low-energy part of the conductivity spectra is 10 times magnified for better visibility. In MnCr${}_2$O${}_4$ and CoCr${}_2$O${}_4$, only the four $T_{1u}\left(\mathrm{IR}\right)$ infrared active phonon modes corresponding to the cubic symmetry are visible. In the other compounds, some of these four modes are clearly split indicating the lowering of the crystal symmetry.

Figure 3

Optical conductivity spectra of MnCr${}_2$O${}_4$ at $T=300$ and 10 K contain four distinct phonon modes. Neither the ferrimagnetic transition at $T_C=51$ K nor the onset of conical spin order at $T_S=14$ K are accompanied with an observable splitting of these $T_{1u}\left(\mathrm{IR}\right)$ phonon modes. Note that the conductivity scales for panels (a) and (b) differ by more than an order of magnitude.

Figure 4

Optical conductivity spectra of CoCr${}_2$O${}_4$ at $T=300$ and 10 K. Similarly to MnCr${}_2$O${}_4$, no splitting of the four $T_{1u}\left(\mathrm{IR}\right)$ phonon modes can be traced either below the ferrimagnetic transition at $T_C=93$ K or upon the conical spin ordering at $T_S=26$ K.

Figure 5

Temperature dependence of phonon mode energies in MnCr${}_2$O${}_4$ and CoCr${}_2$O${}_4$. Mn${}^{2+}$ and Co${}^{2+}$ ions, located on the $A$ site of the spinel structure, have no orbital degeneracy [Fig. 1b]. Correspondingly, no structural transition associated with a cooperative Jahn-Teller distortion has been observed. No measurable splitting of the phonon modes occurs upon the magnetic phase transitions either. $T_C$ and $T_S$ label the transition to a ferrimagnetic state and to the ground state with conical spin order, respectively.

Figure 6

Optical conductivity spectra of FeCr${}_2$O${}_4$ at various temperatures in the vicinity of the Jahn-Teller transition at $T_{\text{JT}}=135$ K, the ferrimagnetic ordering at $T_C=93$ K, and the onset of the conical order at $T_S=35$ K. Lowering of the crystal symmetry can be traced as temperature-induced splitting of the lowest- (a) and the highest-energy (b) phonon modes both at $T_{\text{JT}}$ and $T_C$, while splitting of the other two $T_{1u}\left(\mathrm{IR}\right)$ modes can not be resolved.

Figure 7

(a), (b) Optical conductivity spectra of NiCr${}_2$O${}_2$ at various temperatures over the region of the Jahn-Teller transition at $T_{\text{JT}}=320$ K, the ferrimagnetic ordering at $T$${}_C=74$ K, and the onset of the canted magnetic structure at $T$${}_S=31$ K. Splitting of the third and the fourth $T_{1u}\left(\mathrm{IR}\right)$ modes can be observed both at $T$${}_{\text{JT}}$ and $T$${}_C$. Scale of the low-energy part is magnified 10 times larger. (c) Between these modes, the activation of an $E_u\left(\mathrm{IR}\right)$ mode, which originates from a $T_{2u}\left(\mathrm{S}\right)$ mode being silent in the cubic phase, can be followed. Rigorous analysis shows the activation of another $E_u\left(\mathrm{IR}\right)$ mode upon $T_{\text{JT}}$, which is located among the branches of the third $T_{1u}\left(\mathrm{IR}\right)$ mode. (d) Oscillator strength for this $E_u\left(\mathrm{IR}\right)$ mode and for the branches of the third $T_{1u}\left(\mathrm{IR}\right)$ mode as a function of temperature.

Figure 8

Optical conductivity spectra of CuCr${}_2$O${}_4$ at various temperatures. At room temperature, all $T$${}_{1u}\left(\mathrm{IR}\right)$ phonon modes are split due to the Jahn-Teller transition at $T$${}_{\text{JT}}=854$ K. For each mode, the remaining degeneracy is lifted upon the magnetic ordering at $T_C=152$ K.

Figure 9

Temperature dependence of phonon energies in FeCr${}_2$O${}_4$, NiCr${}_2$O${}_4$, and CuCr${}_2$O${}_4$. Lattice vibrations originating from the same $T_{1u}$ and $T_{2u}$ cubic modes are labeled with the same color and symbol. The temperature of the phase transitions are indicated by dashed lines.

Figure 10

Energy-level diagrams corresponding to Fe${}^{2+}$, Co${}^{2+},$ Ni${}^{2+}$, and Cu${}^{2+}$ 3$d$ orbitals occupied by holes. In these diagrams, only the spin-allowed single-particle excitations are shown, red arrows represent the allowed optical transitions. (a), (b) 3$d$-3$d$ optical transitions within the $d$ band of the $A$-site ion split by the cubic ($T$${}_d$) crystal field $\Delta E$, the Racah parameter $B$, and the spin-orbit interaction $\zeta$ for $A=$ Fe${}^{2+}$ and $A=$ Co${}^{2+}$ (Refs. 26 and 36). (c) In the case of $A=$ Ni${}^{2+}$, the effect of local $T$${}_d$ symmetry and spin-orbit interaction was taken into account, the latter as the weakest perturbation. Here, symmetry lowering associated with the weak tetragonal splitting was neglected. (d) In the case of $A=$ Cu${}^{2+}$, the effect of Jahn-Teller distortion, i.e., tetragonal splitting, $\Delta E_1$ was considered stronger than the spin-orbit interaction.

Figure 11

Analysis of the intra-atomic 3$d$-3$d$ transitions in NiCr${}_2$O${}_4$ and CuCr${}_2$O${}_4$ as shown in panels (a) and (b), respectively. Labels correspond to notation introduced in Fig. 10. Aside from the oscillators, there is a common baseline corresponding to the tail of the band edge.