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We report on our theoretical study of the oscillating magnetoresistance (MR) effect in the spin-polarized transport through a finite carbon nanotube (CNT). We find that the fine structure in MR depends sensitively on the coupling strengths between the ferromagnetic electrodes and a finite CNT. Recent experimental findings can be well explained by our model study. rRecently, a couple of experimental groups [1-3] realized the spin field-effect transistor in the carbon nanotube (CNT) systems. They were able to control the spinpolarized current through a CNT by the third gate electrode which is capacitively coupled to the CNT. The main discoveries of experiments are: (1) the magnetoresistance (MR) is oscillating as a function of the gate voltage, just like the linear conductance; (2) the MR is dipped near the conductance peak and is peaked in the conductance valleys; (3) the MR in some samples can even become negative near the conductance peak. These discoveries can be well explained by the spin-polarized resonant tunneling [4] through a discrete energy levels formed in a finite CNT.In this work we study the spin-polarized transport through a CNT, using the tight-binding Hamiltonian approach to the CNT. Though we adopt the noninteracting electron model in a finite CNT, we are able to reproduce the main features of the experiments. In addition, we show that the MR is dependent sensitively on the coupling strength between the CNT and the two ferromagnetic (FM) electrodes, and on the asymmetry in coupling to two FM electrodes.To study the phase-coherent spin-polarized transport through a finite CNT, we consider the finite armchair-type ðn; nÞ CNT which is end-contacted to the two ferromagnetic electrodes. For the electronic structure of a finite ðn; nÞ CNT with the N layers, we adopt the p-electron tightbinding model which is known to agree well with the ab initio calculations close to the Fermi level, H cnt ¼ P ia g a y ia a ia À t P hi;ji P a ða y ia a ja þ h:c:Þ, where hi; ji denotes the nearest neighbor pairs. The hopping integral is taken to be t ¼ 2:66 eV [5]. The on-site energy g at each carbon site is modulated by the gate voltage. Though the on-site energy may well depend on the separation of each site from the substrate (which is capacitively coupled to the gate electrode), we simply use the uniform energy shift in the on-site energy due to the gate voltage. The FM metals, left (L) and right (R) electrodes, are described by two conduction bands of majority and minority spins. For the end-contact geometry, only the carbon atoms at the left and right edge layers are assumed to be coupled to the FM electrodes. Overall, our model system can be written as H ¼ H cb þ H cnt þ H 1 , where H cb ¼ P p¼L;R P ka pka c y pka c pka , H cnt ¼ P a C y a H cnt C a , and H 1 ¼ P pka ½c y pka V y p ðkÞC a þ h:c:. Here c y pka and c pka are the creation and annihilation operators, respectively, for electrons of wave number k in the electrode p ¼ L; R, with ARTICLE IN PRESS www.elsevier.com/locate/jmmm 0304-8853/$ -see front matte...

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