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Magnetization induced by odd-frequency spin-triplet Cooper pairs in a Josephson junction with metallic trilayers

We theoretically study the magnetization inside a normal metal induced in an s-wave superconductor/ferromagnetic metal/normal metal/ferromagnetic metal/s-wave superconductor (S/F1/N/F2/S) Josephson junction. Using quasiclassical Green's function method, we show that the magnetization becomes finite inside N. The origin of this magnetization is due to odd-frequency spin-triplet Cooper pairs formed by electrons of equal and opposite spins, which are induced by proximity effect in the S/F1/N/F2/S junction. We find that the magnetization M(d,q) in N can be decomposed into two parts, M(d,q)=MI(d)+MII(d,q), where q is the superconducting phase difference between two Ss and d is the thickness of N. MI(d) exists generally in S/F junctions, while MII(d,q) carries all q dependence and represents the fingerprint of phase coherence between two Ss in Josephson junctions. The q dependence thus allows us to control the magnetization in N by tuning q for a fixed d. We show that MI(d) weakly decreases with increasing d, while the q dependent magnetization MII(d,q) rapidly decays with d. Moreover, we find that the time-averaged magnetization <MII(d,q)> exhibits discontinuous peak at each resonance DC voltage Vn=nhw_S/2e(n: integer) when DC voltage V as well as AC voltage v_ac(t) with frequency w_S are both applied to the S/F1/N/F2/S junction. This is because MII(d,q) oscillates generally in time t (AC magnetization) with dq/dt=2e[V+v_ac(t)]/h and thus <MII(d,q)>=0, but can be converted into the time-independent DC magnetization for DC voltage at Vn. We also discuss that the magnetization induced in N can be measurably large in realistic systems. Therefore, the measurement of the induced magnetization serves as an alternative way to detect the phase coherence between two Ss in Josephson junctions. Our results also provide a basic concept for tunable magnetization in superconducting spintronics devices.