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In Mott insulators, band electrons are localized due to strong electron-electron interactions. Although the s-electrons of alkali metals are very delocalized, by confining them in the periodic nanospace of zeolite crystals and making them moderately localized, such a strongly correlated electron system can be created.$^1$ In sodalite, $\beta$-cages with an inner diameter of 0.7 nm are arranged in a bcc structure. By loading alkali atoms, an $A_4^{3+}$ cluster ($A$: alkali atom) is formed in the cage. The cluster has one unpaired s-electron. Antiferromagnetic order of Mott insulating state has been identified in $A =$ Na, K, and K-Rb alloy clusters.$^2$ $T_N$ systematically increases from 50 K (Na) to 90 K (K-Rb alloy). In ZF-$\mu^+$SR, a uniform local field is observed below $T_N$, and its value is higher for clusters with heavier chemical compositions.$^2$
To clarify the mechanism of the systematic change in the local field and its relation with the Mott-insulating state of this system, we investigate the muon Knight shift by high TF-$\mu^+$SR using NuTime at TRIUMF. We successfully obtained the hyperfine coupling constants between $\mu^+$ and the s-electron above $T_N$ from the $K-\chi$ plot. By combining the ZF-$\mu^+$SR local field,$^2$ we determined the size of the ordered moments, which systematically decreases from $\simeq$ 0.5 $\mu_B$ (Na) to $\simeq$ 0.3 $\mu_B$ (K-Rb alloy). It correlates perfectly with the increase in $T_N$, namely, the decrease in the electron correlation $U/t$ in the Mott-Hubbard model. From DFT calculations, we found that $\mu^+$ is in a hydride (Mu−) state at the cage center. This also explains that the systematic increase in the local field corresponds to the decrease in $U/t$ due to the shallower potential of the heavier alkali atoms.
$^1$T. Nakano and Y. Nozue, Adv. Phys.: X 2, 254-280 (2017).
$^2$T. Nakano et al., J. Phys. Soc. Jpn. 79, 073707-1-4 (2010).
