Spin excitations in frustrated $A$-site spinels investigated with inelastic neutron scattering

The magnetic ions of the $A$-site spinels $M{\text{Al}}_2{\text{O}}_4\left(M=\text{Mn},\text{Fe},\text{Co}\right)$ form a diamond lattice which, by means of competing interactions, are subject to strong frustration effects. In this paper we present the outcome of inelastic neutron-scattering experiments to investigate the corresponding magnetic excitations. ${\text{MnAl}}_2{\text{O}}_4$ exhibits the strongest magnetic exchange and reveals long-range magnetic order below $T_N=40\text{K}$ with a well-defined spin-wave dispersion. In contrast, ${\text{CoAl}}_2{\text{O}}_4$ and ${\text{FeAl}}_2{\text{O}}_4$ do not show long-range order but the magnetic scattering corresponds to a liquidlike structure factor. Inelastic neutron scattering in ${\text{CoAl}}_2{\text{O}}_4$ shows a dispersionlike structure indicating collective magnetic excitations, despite the absence of long-range magnetic order. This type of behavior corresponds to a cooperative paramagnet. The opening of a spin gap in ${\text{FeAl}}_2\text{O}4$ is observed at low temperatures. These three $A$-site spinels are located on the border of long-range magnetic order with distinct emergent behavior. The corresponding role of orbital degrees of freedom is discussed.

Figure 1

(Color online) Contour plot of the dynamic structure factor $S\left(Q,\omega \right)$ of ${\text{MnAl}}_2{\text{O}}_4$ for various temperatures. Magnetic intensities are shown by a suitable color coding on a logarithmic scale (darker grey upon increasing intensity up to the background level at high energy transfers). The solid line in the lowest frame shows the calculated spin-wave dispersion of ${\text{MnAl}}_2{\text{O}}_4$ along $\left(\xi ,\xi ,\xi \right)$ including nearest- and next-nearest-neighbor ions as described in the text. The Bragg-peak positions of the (111) reflection at $1.33{\text{Å}}^{-1}$ and the (222) reflection at $2.65{\text{Å}}^{-1}$ are indicated.

Figure 2

(Color online) Calculated normalized spin-wave dispersion of ${\text{MnAl}}_2{\text{O}}_4$ for nearest-neighbor exchange along the three principal cubic directions. Details are described in the text.

Figure 3

(Color online) Neutron-scattering spectra at constant wave-vector transfer $Q=1.25{\text{Å}}^{-1}$ of ${\text{MnAl}}_2{\text{O}}_4$ for various temperatures. The blue dashed lines correspond to the elastic line as measured by a vanadium standard, the individual fit components are drawn as black lines and the resulting fit is represented by the full red line.

Figure 4

(Color online) Linewidths and eigenfrequencies of ${\text{MnAl}}_2{\text{O}}_4$ at $Q=1.25{\text{Å}}^{-1}$ versus temperature. Full circles represent the quasielastic linewidth $\Gamma$ and open symbols show the excitation energy ${\omega }_j$ and linewidth ${\Gamma }_j$ of the inelastic damped harmonic oscillator at low temperatures. The Néel temperature is indicated by the vertical dashed line. Full lines are drawn to guide the eyes.

Figure 5

(Color online) $Q$ dependence of the integrated quasielastic scattering intensities of ${\text{MnAl}}_2{\text{O}}_4$ at 160 K (open red circles) and 100 K (black circles) in the paramagnetic state as resulting from the fits. The solid line corresponds to single-ion behavior according to the neutron magnetic form factor within the dipole approximation.

Figure 6

(Color online) Contour plot of the dynamic structure factor $S\left(Q,\omega \right)$ of ${\text{CoAl}}_2{\text{O}}_4$ for various temperatures. Magnetic intensities are shown by a suitable color coding on a logarithmic scale.

Figure 7

(Color online) Neutron-scattering spectra of ${\text{CoAl}}_2{\text{O}}_4$ at constant wave-vector transfer $Q=1.25{\text{Å}}^{-1}$ for various temperatures. The blue dashed lines correspond to the elastic scattering as measured by a vanadium standard, the individual fit components are drawn as black lines and the resulting fit is represented by the full red line.

Figure 8

(Color online) Linewidth and eigenfrequencies of ${\text{CoAl}}_2{\text{O}}_4$ at $Q=1.25{\text{Å}}^{-1}$ versus temperature. Full circles represent the quasielastic linewidth and open symbols show the excitation energy (triangle) and linewidth (circles) of the inelastic damped harmonic oscillator at low temperatures. The characteristic temperature $T^{\ast }$ is indicated. The full lines are drawn to guide the eyes.

Figure 9

(Color online) Contour plot of the dynamic structure factor $S\left(Q,\omega \right)$ of ${\text{FeAl}}_2{\text{O}}_4$ for various temperatures. Magnetic intensities are shown by a suitable color coding on a logarithmic scale. A band of magnetic excitations with a mean energy of 4 meV without dispersion is indicated by the dashed line.

Figure 10

(Color online) Structure factor $S\left(\omega \right)$ at constant wave vector transfer $Q=1.25{\text{Å}}^{-1}$ of ${\text{FeAl}}_2{\text{O}}_4$ for various temperatures. The blue dashed lines correspond to the elastic line as measured by a vanadium standard, the individual fit components are drawn as black lines and the resulting fit is represented by the full red line. The lowest frame shows a simultaneous plot of ${\chi }^{″}\left(\omega \right)$ for four different temperatures to evidence the evolution of a spin gap.

Figure 11

(Color online) Linewidth and eigenfrequencies of ${\text{FeAl}}_2{\text{O}}_4$ at $Q=1.25{\text{Å}}^{-1}$ versus temperature. Full circles represent the quasielastic linewidth and open symbols show the excitation energy (triangles) and linewidth (circles) of the inelastic damped harmonic oscillator at low temperatures. The characteristic temperature $T^{\ast }$ is indicated. Full lines are drawn to guide the eyes.

Figure 12

(Color online) High-resolution spectrum of ${\text{FeAl}}_2{\text{O}}_4$ at $T=1.8\text{K}$ revealing an energy pseudogap for magnetic excitations. The dynamic structure factor (full red line) is the sum of a quasielastic and an inelastic contribution (thin black lines). It is the result of a cut in the low-energy or low-$Q$ region of $S\left(Q,\omega \right)$ as shown in the inset at constant $Q=1.25{\text{Å}}^{-1}$ (vertical dashed white line).