Antiferromagnetic resonance in alkali-metal clusters in sodalite

We have performed electron spin resonance (ESR) studies of K${}_4^{3+}$ and (K${}_3$Rb)${}^{3+}$ nanoclusters incorporated in powder specimens of aluminosilicate sodalite at several microwave frequencies between 9 and 34 GHz. The K${}_4^{3+}$ and (K${}_3$Rb)${}^{3+}$ clusters are arrayed in a bcc structure and are known to show antiferromagnetic ordering below the N$\acute{\mathrm{e}}$el temperatures of $T_{\mathrm{N}}$ $\simeq 72$ and $\simeq$80 K, respectively, due to the exchange coupling between $s$ electrons confined in the clusters. We have found sudden broadenings of ESR spectra in both samples below $T_{\mathrm{N}}$. The line shape of the spectra below $T_{\mathrm{N}}$ is analyzed by powder pattern simulations of antiferromagnetic resonance (AFMR) spectra. The calculated line shapes well reproduce the experimental ones at all the frequencies by assuming a biaxial magnetic anisotropy. We have evaluated extremely small anisotropy fields of approximately 1 Oe indicating that these materials are ideal Heisenberg antiferromagnets. We have also found that the magnetic anisotropy changes from easy-plane type to uniaxial type by changing into a heavier alkali-metal cluster and that the $g$ value shifts to a large value beyond two below $T_{\mathrm{N}}$ for K${}_4^{3+}$ and (K${}_3$Rb)${}^{3+}$ nanoclusters. These novel features of K${}_4^{3+}$ and (K${}_3$Rb)${}^{3+}$ nanoclusters incorporated in sodalite are discussed.

Figure 1

Schematic illustrations of (a) crystal structure of sodalite and (b) $A_4^{3+}$ cluster formed in the $\beta$ cage, where $A$ is an alkali atom. The framework has the SOD-type structure with a bcc arrangement of the $\beta$ cages.

Figure 2

Temperature dependence of ESR spectra of K${}_4^{3+}$ clusters in sodalite measured at an X-band frequency, 9.31 GHz.

Figure 3

Temperature dependence of ESR spectra of (K${}_3$Rb)${}^{3+}$ clusters in sodalite measured at X-band frequency, 9.29 GHz.

Figure 4

Temperature dependence of the ESR linewidth (FWHM) for Na${}_4^{3+}$, K${}_4^{3+}$ and (K${}_3$Rb)${}^{3+}$ clusters in sodalite measured at X-band frequency. The width is plotted on a logarithmic scale. The data for Na${}_4^{3+}$ clusters are taken from Ref. 10.

Figure 5

ESR spectra for Na${}_4^{3+}$, K${}_4^{3+}$ and (K${}_3$Rb)${}^{3+}$ clusters in sodalite measured at designated temperatures and at X-band frequencies. The data for Na${}_4^{3+}$ clusters are from Ref. 10. The short vertical bars indicate resonance fields at $g=2$ for each frequency.

Figure 6

ESR spectra for K${}_4^{3+}$ and (K${}_3$Rb)${}^{3+}$ clusters in sodalite measured at liquid helium temperatures at several frequencies. The black open circles with a solid line show the experimental data. The red solid curves and the blue dashed curves are the simulation results of the AFMR powder pattern by assuming an isotropic and an anisotropic relaxation processes, respectively.

Figure 7

Resonance modes of a biaxial antiferromagnet at absolute zero. The red solid, green broken, and blue dotted curves are those for magnetic fields along the hard axis, the second-easy axis and the easy axis, respectively. The dashed-and-dotted line shows a paramagnetic resonance mode with $\omega /\gamma =H$. The zero-field gaps ${\nu }_1$ and ${\nu }_2$ are given by $2\pi {\nu }_1/\gamma =\sqrt{2H_{\mathrm{E}}{H}_{\mathrm{A}1}}$ and $2\pi {\nu }_2/\gamma =\sqrt{2H_{\mathrm{E}}{H}_{\mathrm{A}2}}$, respectively. The spin-flop field $H_{\mathrm{sf}}$ is given by $H_{\mathrm{sf}}=\sqrt{2H_{\mathrm{E}}{H}_{\mathrm{A}1}}$.

Figure 8

Temperature dependencies of anisotropy field and g value for Na${}_4^{3+}$, K${}_4^{3+}$ and (K${}_3$Rb)${}^{3+}$ clusters in sodalite. For the anisotropy field, $\sqrt{H_{\mathrm{A}2}}$ is plotted. The data for Na${}_4^{3+}$ clusters are taken from Ref. 10.

Figure 9

Alkali-element dependencies of anisotropy field and $g$ value for Na${}_4^{3+}$, K${}_4^{3+}$ and (K${}_3$Rb)${}^{3+}$ clusters in sodalite. The data for Na${}_4^{3+}$ clusters are taken from Ref. 10. The horizontal dotted line in the lower panel indicates the free electron $g$ value ($g_{\mathrm{e}}=2.0023$).