  In this note, we classify biharmonic submanifolds in a sphere defined by
bi-eigenmaps ($\Delta^2 \phi=\lambda \phi$) or buckling eigenmaps ($\Delta^2
\phi=-\mu \Delta \phi$). We then classify biharmonic bi-eigenmaps and buckling
eigenmaps into spheres with constant energy density. The results can be viewed
as generalizations of Takahashi's characterization of minimal submanifolds in a
sphere by eigenmaps.
