  Imbalance of Mn3+ and Mn4+ ions between surface and bulk in layered
manganites is investigated on the basis of an orbital degenerate
double-exchange model with surfaces. For two types of t2g spin structures, the
number of Mn4+ ions on the surface is found to be larger than that in the bulk,
explained by the surface-induced Friedel oscillation. In particular, for the
sheet-type antiferromagnetic state, quantum beat phenomenon appears due to the
interference among Friedel oscillations and the number of Mn4+ ions tends to
increase on the surface. The present results are believed to be useful to
develop high-efficient cathodes in Li-ion batteries and catalysts.
