Universal Exchange-Driven Phonon Splitting in Antiferromagnets

We report a linear dependence of the phonon splitting $\Delta \omega$ on the nondominant exchange coupling constant $J_{\mathrm{nd}}$ in the antiferromagnetic transition-metal monoxides MnO, FeO, CoO, NiO, and in the frustrated antiferromagnetic oxide spinels ${\mathrm{CdCr}}_2{\mathrm{O}}_4$, ${\mathrm{MgCr}}_2{\mathrm{O}}_4$, and ${\mathrm{ZnCr}}_2{\mathrm{O}}_4$. It directly confirms the theoretical prediction of an exchange-induced splitting of the zone-center optical phonon for the monoxides and explains the magnitude and the change of sign of the phonon splitting on changing the sign of the nondominant exchange also in the frustrated oxide spinels. The experimentally found linear relation $\hslash \Delta \omega =\beta J_{\mathrm{nd}}{S}^2$ with slope $\beta =3.7$ describes the splitting for both systems and agrees with the observations in the antiferromagnets ${\mathrm{KCoF}}_3$ and ${\mathrm{KNiF}}_3$ with perovskite structure and negligible next-nearest neighbor coupling. The common behavior found for very different classes of cubic antiferromagnets suggests a universal dependence of the exchange-induced phonon splitting at the antiferromagnetic transition on the nondominant exchange coupling.

Figure 1

Magnetic properties of the investigated TMMOs. (a) Magnetic unit cell showing the antiferromagnetic order along [111]. (b) Splitting into phonon modes with eigenfrequencies ${\omega }_{\parallel }$ and ${\omega }_{\perp }$. (c) Nearest-neighbor coupling $J_1$ and dominant next-nearest neighbor coupling $J_2$. (d) Neel temperatures vs $J_{2}S(S+1)$ using values for $J_2$ taken from Ref. 26 in comparison to the mean-field expectation (dashed line).

Figure 2

(a) Cubic spinel structure of ${\mathrm{ACr}}_2{\mathrm{O}}_4$. Nonmagnetic $A$-site ions are in tetrahedral and the Cr ions in octahedral environment. (b) Nearest-neighbor and effective further nearest-neighbor exchange paths of the Cr ions on the pyrochlore lattice [30, 31]. (c) Néel temperatures vs $J_{\mathrm{nn}}S(S+1)$ using values for $J_{\mathrm{nn}}$ taken from Ref. 30 in comparison to the mean-field expectation (dashed line).

Figure 3

Dielectric loss of (a) MnO, (b) ${\mathrm{Fe}}_{0.92}\mathrm{O}$, (c) CoO, (d) NiO, (e) ${\mathrm{CdCr}}_2{\mathrm{O}}_4$, (f) ${\mathrm{ZnCr}}_2{\mathrm{O}}_4$, and (g) ${\mathrm{MgCr}}_2{\mathrm{O}}_4$ around the TO phonon modes above and below, $T_N$ respectively. The high temperature data of ${\mathrm{Fe}}_{0.92}\mathrm{O}$ and ${\mathrm{MgCr}}_2{\mathrm{O}}_4$ are shifted upward for clarity. Lines indicate fits to the loss peaks (see text).

Figure 4

Phonon splitting $\hslash \Delta \omega$ vs nondominant exchange contributions $J_1{S}^2$ and $J_{\mathrm{nnn}}{S}^2$ for the investigated TMMOs and Cr spinels. The solid line is a linear fit to the experimental data of the TMMOs. The values for $J_1$ and $J_{\mathrm{nnn}}$ are taken from Refs. 26, 30, respectively. Values for the perovskite fluorides were taken from Refs. 48, 49.