  We analytically study the magnetic response of persistent current (PC) in
normally non-interacting mesoscopic rings of bimodal potential with nearest
neighboring interactions (t) and alternating site energies. It is shown that a
ring of perimeter (N) and width (M) generally shows weak diamagnetic, breaking
the even-odd rule of electron filling. Especially, a maximal paramagnetic
current in primary F0/2 period is predicted at N=(2p+1)(M+1) with odd M and
integer p, while a maximal diamagnetic F0/2- current obtained at
N=(2p+1)(M+1)+/-1 with even M. The current amplitudes depend strongly on both N
and M, varied by at least 1~2 orders of magnitude, exhibiting a remarkable
quantum size effect. A current limit of paramagnetic harmonics is expected at
N=2p(M+1), independent of the sizes of N and M, in favor of experiment
observation. A new mechanism of magnetic response is proposed that an electron
circling the ring shall pass successively each channel within one flux quantum,
accumulating an additional phase on each inter-channel transition, which leads
to the paramagnetic-diamagnetic transition and period halving. The results
unify and unveil the contradictions in PC between theory and experiments,
validating quantum mechanics at mesoscopic scale.
